THE  LIBRARY 
OF 

THE  UNIVERSITY 
OF  CALIFORNIA 


GIFT  OF 


Prof.  G.  C.  Evans 


mm 


Linear  Associative  Algebra. 


By  BENJAMIN  PEIRCE,  LL.  D. 

I  \ 

LATE  PERKINS  PROFESSOR  OF  ASTRONOMY  AND  MATHEMATICS  IN  HARVARD  UNIVERSITY 
AND  SUPERINTENDENT  OF  THE  UNITED  STATES  COAST  SURVEY. 


New  Edition,  with  Addenda  and  Notes,  by  C.  S.  PEIRCE,  Son  of  the  Author. 


[Extracted  from  The  American  Journal  of  Mathematics^ 


NEW  YORK  :    D.  VAN  NOSTRAND,  PUBLISHER. 

1882. 


PRESS  OF  ISAAC  FRIBDBNWALD, 
Baltimore,  Md. 


ERRATA. 

Page  10,  §  31.     The  first  formula  should  read 

(A±B)  C=AC±BC. 
Page  30.     The  third  formula  should  read 

k(i  —  k)=j. 
Page  36.     Foot-note,  second  line  of  second  paragraph,  read 

J=~(h  —  ^),    ^-^(l+jy. 

Page  40.     Last  line  of  foot-note.     For  e ,  read  I . 

Page  52.     Multiplication  table  of  ft) .     For  ji  =  i,  read  ji=j. 

Page  75.     Last  line  of  foot-note,  insert  I,  at  beginning  of  line. 

Page  86.     Foot-note.     Add  that  on  substituting  k  +  vj  for  k ,  the  algebra 
)  reduces  to  (oarB) ;  and  the  same  substitution  reduces  (ays)  to  (agB) . 

Page  91.     Last  line  of  foot-note.     For  i,  read  I. 


PREFACE. 

Lithographed  copies  of  this  book  were  distributed  by  Professor  Peirce  among  his 
friends  in  1870.  The  present  issue  consists  of  separate  copies  extracted  from  The  Ameri 
can  Journal  of  Mathematics,  where  the  work  has  at  length  been  published.* 

The  body  of  the  text  has  been  printed  directly  from  the  lithograph  with  only  slight 
verbal  changes.  Appended  to  it  will  be  found  a  reprint  of  a  paper  by  Professor  Peirce, 
dated  1875,  and  two  brief  contributions  by  the  editor.  The  foot-notes  contain  transforma- 

* 

tions  of  several  of  the  algebras,  as  well  as  what  appeared  necessary  in  order  to  complete 
the  analysis  in  the  text  at  a  few  points.    A  relative  form  is  also  given  for  each  algebra  ;  for 
the  rule  in  Addendum  II.  by  which  such  forms  may  be  immediately  written  down,  was 
unknown  until  the  printing  was  approaching  completion. 
The  original  edition  was  prefaced  by  this  dedication  : 

To  MY  FEIENDS. 

This  work  has  been  the  pleasantest  mathematical  effort  of  my  life.  In  no  other  have 
I  seemed  to  myself  to  have  received  so  full  a  reward  for  my  mental  labor  in  the  novelty 
and  breadth  of  the  results.  I  presume  that  to  the  uninitiated  the  formulae  will  appear  cold 
and  cheerless  ;  but  let  it  be  remembered  that,  like  other  mathematical  formulae,  they  find 
their  origin  in  the  divine  source  of  all  geometry.  Whether  1  shall  have  the  satisfaction  of 
taking  part  in  their  exposition,  or  whether  that  will  remain  for  some  more  profound 
expositor,  will  be  seen  in  the  future. 

B.  P. 

*  To  page  n  of  this  issue  corresponds  page  »+96  of  Vol.  IV.  of  The  JoumaT~ 


342 


LINEAR  ASSOCIATIVE  ALGEBRA. 


1.  Mathematics  is  the  science  which  draws  necessary  conclusions. 

This  definition  of  mathematics  is  wider  than  that  which  is  ordinarily  given, 
and  by  which  its  range  is  limited  to  quantitative  research.  The  ordinary 
definition,  like  those  of  other  sciences,  is  objective  ;  whereas  this  is  subjective. 
Recent  investigations,  of  which  quaternions  is  the  most  noteworthy  instance, 
make  it  manifest  that  the  old  definition  is  too  restricted.  The  sphere  of  mathe 
matics  is  here  extended,  in  accordance  with  the  derivation  of  its  name,  to  all 
demonstrative  research,  so  as  to  include  all  knowledge  strictly  capable  of  dog 
matic  teaching.  Mathematics  is  not  the  discoverer  of  laws,  for  it  is  not 
induction  ;  neither  is  it  the  framer  of  theories,  for  it  is  not  hypothesis ;  but  it  is 
the  judge  over  both,  and  it  is  the  arbiter  to  which  each  must  refer  its  claims ; 
and  neither  law  can  rule  nor  theory  explain  without  the  sanction  of  mathematics. 
It  deduces  from  a  law  all  its  consequences,  and  develops  them  into  the  suitable 
form  for  comparison  with  observation,  and  thereby  measures  the  strength  of  the 
argument  from  observation  in  favor  of  a  proposed  law  or  of  a  proposed  form  of 
application  of  a  law. 

Mathematics,  under  this  definition,  belongs  to  every  enquiry,  moral  as  well 
as  physical.  Even  the  rules  of  logic,  by  which  it  is  rigidly  bound,  could  not  be 
deduced  without  its  aid.  The  laws  of  argument  admit  of  simple  statement,  but 
they  must  be  curiously  transposed  before  they  can  be  applied  to  the  living  speech 
and  verified  by,observation.  In  its  pure  and  simple  form  the  syllogism  cannot 
be  directly  compared  with  all  experience,  or  it  would  not  have  required  an 


2  PEIRCE  :    Linear  Associative  Algebra. 

Aristotle  to  discover  it.  It  must  be  transmuted  into  all  the  possible  shapes  in 
which  reasoning  loves  to  clothe  itself.  The  transmutation  is  the  mathematical 
process  in  the  establishment  of  the  law.  Of  some  sciences,  it  is  so  large  a 
portion  that  they  have  been  quite  abandoned  to  the  mathematician, — which 
may  not  have  been  altogether  to  the  advantage  of  philosophy.  Such  is  the 
case  with  geometry  and  analytic  mechanics.  But  in  many  other  sciences,  as  in 
all  those  of  mental  philosophy  and  most  of  the  branches  of  natural  history,  the 
deductions  are  so  immediate  and  of  such  simple  construction,  that  it  is  of  no 
practical  use  to  separate  the  mathematical  portion  and  subject  it  to  isolated 
discussion. 

2.  The  branches  of  mathematics  are  as  various  as  the  sciences  to  which  they 
belong,  and  each  subject  of  physical  enquiry  has  its  appropriate  mathematics. 
In  every  form  of  material  manifestation,  there  is  a  corresponding  form  of  human 
thought,  so  that  the  human  mind  is  as  wide  in  its  range  of  thought  as  the 
physical  universe  in  which  it  thinks.     The  two  are  wonderfully  matched.     But 
where  there  is  a  great  diversity  of  physical  appearance,  there  is  often  a  close 
resemblance  in  the  processes  of  deduction.     It  is  important,  therefore,  to  separate 
the  intellectual. work  from  the  external  form.     Symbols  must  be  adopted  which 
may  serve  for  the  embodiment  of  forms  of  argument,  without  being  trammeled 
by    the    conditions   of  external  representation  or  special  interpretation.     The 
words   of  common  language  are  usually  unfit  for  this  purpose,  so  that  other 
symbols  must  be  adopted,  and  mathematics  treated  by  such  symbols  is  called 
algebra.     Algebra,  then,  is  formal  mathematics. 

3.  All  relations  are  either  qualitative  or  quantitative.    Qualitative  relations 
can  be  considered  by  themselves  without  regard  to  quantity.     The  algebra  of 
such  enquiries  may  be  called  logical  algebra,  of  which  a  fine  example  is  given 
by  Boole. 

Quantitative  relations  may  also  be  considered  by  themselves  without  regard 
to  quality.  They  belong  to  arithmetic,  and  the  corresponding  algebra  is  the 
common  or  arithmetical  algebra. 

In  all  other  algebras  both  relations  must  be  combined,  and  the  algebra  must 
conform  to  the  character  of  the  relations. 

4.  The  symbols  of  an  algebra,  with  the  laws  of  combination,  constitute  its 
language ;  the  methods  of  using  the  symbols  in  the  drawing  of  inferences  is  its 
art;  and  their  interpretation  is  its  scientific  application.     This  three-fold  analysis 
of  algebra  is  adopted  from  President  Hill,  of  Harvard  University,  and  is  made 
the  basis  of  a  division  into  books. 


PEIRCE  :    Linear  Associative  Algebra.  3 

BOOK  I.* 
THE  LANGUAGE  OF  ALGEBRA. 

5.  The  language  of  algebra  has  its  alphabet,  vocabulary,  and  grammar. 

6.  The  symbols  of   algebra  are    of   two    kinds :    one    class    represent    its 
fundamental  conceptions  and  may  be  called  its  letters,  and  the  other  represent 
the  relations  or  modes  of  combination  of  the  letters  and  are  called  the  signs. 

7.  The  alphabet  of  an  algebra  consists  of  its  letters ;  the  vocabulary  defines 
its  signs  and  the  elementary  combinations  of  its  letters ;  and  the  grammar  gives 
the  rules  of  composition  by  which  the  letters  and  signs   are    united    into    a 
complete  and  consistent  system. 

The  Alphabet. 

8.  Algebras  may  be  distinguished  from  each  other  by  the  number  of  their 
independent  fundamental  conceptions,  or  of  the  letters  of  their  alphabet.     Thus 
an  algebra  which  has  only  one  letter  in  its  alphabet  is  a  single  algebra ;  one 
which  has  two  letters  is  a  double  algebra ;  one  of  three  letters  a  triple  algebra ; 
one  of  four  letters  a  quadruple  algebra,  and  so  on. 

This  artificial  division  of  the  algebras  is  cold  and  uninstructive  like  the 
artificial  Linnean  system  of  botany.  But  it  is  useful  in  a  preliminary  investiga 
tion  of  algebras,  until  a  sufficient  variety  is  obtained  to  afford  the  material  for  a 
natural  classification. 

Each  fundamental  conception  may  be  called  a  unit;  and  thus  each  unit  has 
its  corresponding  letter,  and  the  two  words,  unit  and  letter,  may  often  be  used 
indiscriminately  in  place  of  each  other,  when  it  cannot  cause  confusion. 

9.  The  present  investigation,  not  usually  extending  beyond  the  sextuple 
algebra,  limits  the  demand  of  the  algebra  for  the  most  part  to  six  letters ;  and 
the  six  letters,  i,  j,  k,  I,  m  and  n,  will  be  restricted  to  this  use  except  in 
special  cases. 

10.  For  any  given  letter  another  may  be  substituted,  provided  a  new  letter 
represents  a  combination  of  the  original  letters  of  which  the  replaced  letter  is  a 
necessary  component. 

For  example,  any  combination  of  two  letters,  which  is  entirely  dependent 
for  its  value  upon  both  of  its  components,  such  as  their  sum,  difference,  or 
product,  may  be  substituted  for  either  of  them. 


Only  this  book  was  ever  written.     [C.  S.  P.] 


of  letter*  is  radically  important,  and  is  a 

of  originality  in  the  present  investigation:  and  without  it.  such 
investigation  would  have  been  impossible.  It  enables  the  geometer  to 
analyse  am  algebra,  reduce  h  to  its  simplest  and  characteristic  forms,  and 
compare  it  witk  offer  algebras.  It  involves  in  its  principle  a  corresponding 
substitution  of  wok  of  which  it  is  in  reality  the  formal  representative. 

There  is.  however,  no  danger  in  working  with  Hie  symbols,  irrespective  of 
the  t4r?t*  allai**^  to  Ihr  «,  and  the  consideration  of  die  change  of  the  original 
be  safety  reserved  for  the  boat  of  tmkrpntatitm. 

11.  In  making  die  substitution  of  letters,  the  original  letter  wffl  be  preserved 
with  the  distinction  of  a  subscript  number. 

That,  for  the  letter  t  there  may  successively  be  substituted  h,  s,,  s,7  etc.  In 
the  fe^l  forms,  the  sobscript  numbers  can  be  omitted,  and  the y  may  be  omitted 
at  any  period  of  the  investigation,  when  it  will  not  produce  confusion. 

It  win  be  practically  found  that  these  subscript  numbers  need  scarcely  ever 
be  written.  They  pass  through  the  mind,  as  a  sure  ideal  protection  from  erro 
neous  substitution,  but  disappear  from  the  writing  with  the  same  fecility  with 
which  those  evanescent  chemical  compounds,  which  are  essential  to  the  theory 
of  tnmsformation,  escape  the  eye  of  the  observer. 

12.  A  JHOK  algebra  is  one  in  which  every  letter  is  connected  by  some 
indissoluble  relation  with  every  other  letter. 

13.  When  the  letters  of  an  algebra  can  be  separated  into  two  groups,  which 
are  mutually  independent,  it  is  a  sttzx*?  oJ.yJjrfj.     It  is  mixed  even  when  there 
are  letters  common  to  the  two  groups,  provided  those  which  are  not  common  to 
the  two  groups  are  mutually  independent.    Were  an  algebra  employed  for  the 
simultaneous  discussion  of  distinct  classes  of  phenomena,,  such  as  those  of  sound 
and  light,  and  were  the  peculiar  units  of  each  class  to  have  their  appropriate 
letters,  but  were  there  no  recognized  dependence  of  the  phenomena  upon  each 
other,  so  that  the  phenomena  of  each  class  might  have  been  submitted  to 
independent  research,  the  one  algebra  would  be  actually  a  mixture  of  two 
algebras,  one  appropriate  to  sound,  the  other  to  light 

It  may  be  farther  observed  that  when,  in  such  a  case  as  this,  the  component 
algebras  are  identical  in  form,  they  are  reduced  to  the  case  of  one  algebra  with 
two  diverse  interpretations. 


PEIRCE  :    Linear   A*-*jci4tice  Algebrn.  5 

The  Vocabulary. 

14.  Letters   which  are  not  appropriated  to  the  alphabet  of  the  algebra  * 
may  be  used  in  any  convenient  sense.     But  it  is  well  to  employ  the  *mafl  letters 
for  expressions  of  common  algebra,  and  the  capital  letters  for  those  of  the  algebra 
under  discussion. 

There  must,  however,  be  exceptions  to  this  notation :  thus  the  letter  D  will 
denote  the  derivative  of  an  expression  to  which  it  is  applied,  and  ^  the  summa 
tion  of  cognate  expressions,  and  other  exceptions  will  be  mentioned  as  they 
occur.  Greek  letters  will  generally  be  reserved  for  angular  and  functional 
notation. 

15.  The  three  symbols  J  .  d .  and  (3  will  be  adopted  with  the  signification 


J=  V—  1 

9  =  the  ratio  of  circumference  to  diameter  of  circle  =  3.l4l592t  ~ 
6  =  the  base  of  Xaperian  logarithms  =  2.7182818285, 

which  gives  the  mysterious  formula 

J—  '  =  v/  6  9  =4.810477381. 

16.  All  the  signs  of  common  algebra  will  be  adopted:   but  any  signification 
will  be  permitted  them  which  is  not   inconsistent  with  their  use  in  common 
algebra  :  so  that,  if  by  any  process  an  expression  to  which  they  refer  is  reduced 
to  one  of  common  algebra,  they  must  resume  their  ordinary  signification. 

17.  The  sign  =.  which  is  called  that  of  equality,  is  used  in  its  ordinary  sense 
to  denote  that   the   two   expressions  which  it  separates  are  the  same  whole. 
although  they  represent  different  combinations  of  parts. 

18.  The  signs  >  and  <  which  are  those  of  inequality,  and  denote  "more 
than  "  or  "  less  than  "  in  quantity,  will  be  used  to  denote  the  relations  of  a  whole 
to  its  part,  so  that  the  symbol  which  denotes  the  pan  shall  be  at  the  vertex  of 
the  angle,  and  that  which  denotes  the  whole  at  its  opening.     This  involves  the 
proposition  that  the  smaller  of  the  quantities  is  included  in  the  class  expre> 

by  the  larger.     Thus 

B<A  or  A>B 

denotes  that  .4  is  a  whole  of  which  B  is  a  part,  so  that  all  B  is  -4> 


formula  in  the  text  implies,  also,  that  some  A  is  not  B.    [C.  S.  P.] 


t>  PEIRCE  :    Linear  Associative  Algebra. 

If  the  usual  algebra  had  originated  in  qualitative,  instead  of  quantitative, 
investigations,  the  use  of  the  symbols  might  easily  have  been  reversed  ;  for  it 
seems  that  all  conceptions  involved  in  A  must  also  be  involved  in  B  ,  so  that  B 
is  more  than  A  in  the  sense  that  it  involves  more  ideas. 

The  combined  expression 

5>  G<A 

denotes  that  there  are  quantities  expressed  by  G  which  belong  to  the  class  A 
and  also  to  the  class  B.  It  implies,  therefore,  that  some  B  is  A  and  that  some  A  is 
B*  The  intermediate  G  might  be  omitted  if  this  were  the  only  proposition 
intended  to  be  expressed,  and  we  might  write 


In  like  manner  the  combined  expression 

B  <  G  >  A 

denotes  that  there  is  a  class  which  includes  both  A  and  B,-\-  which  proposition 
might  be  written 

B<>A. 

19.  A  vertical  mark  drawn  through  either  of  the  preceding  signs  reverses  its 
signification.     Thus 

A^-  B 

denotes  that  B  and  A  are  essentially  different  wholes  ; 

A^>B   or   B^A 

denotes  that  all  B  is  not  A  ,  J  so  that  if  they  have  only  quantitative  relations, 
they  must  bear  to  each  other  the  relation  of 

A  =  B  or  A  <  B  . 

20.  The  sign  -|-  is  called  plus  in  common  algebra  and  denotes  addition.     'It 
may  be  retained  with  the  same  name,  and  the  process  which  it  indicates  may  be 
called  addition.     In  the  simplest  cases  it  expresses  a  mere  mixture,  in  which 

*  This,  of  course,  supposes  that  Cdoes  not  vanish.     [C.  S.  P.] 
t  The  universe  will  be  such  a  class  unless  A  or  B  is  the  universe.     [C.  S.  P.] 

I  The  general  interpretation  is  rather  that  either  A  and  B  are  identical  or  that  some  B  is  not  A. 
[C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra.  7 

the  elements  preserve  their  mutual  independence.  If  the  elements  cannot  be 
mixed  without  mutual  action  and  a  consequent  change  of  constitution,  the  mere 
union  is  still  expressed  by  the  sign  of  addition,  although  some  other  symbol  is 
required  to  express  the  character  of  the  mixture  as  a  peculiar  compound  having 
properties  different  from  its  elements.  It  is  obvious  from  the  simplicity  of  the 
union  recognized  in  this  sign,  that  the  order  of  the  admixture  of  the  elements 
cannot  affect  it  ;  so  that  it  may  be  assumed  that 

A-}-  B  =  B  +  A 
and 


21.  The  sign  —  is  called  minus  in  common  algebra,  and  denotes  subtraction. 
Retaining  the   same  name,   the   process  is  to  be  regarded  as  the   reverse   of 
addition  ;   so  that  if  an  expression  is  first  added  and  then  subtracted,  or  the 
reverse,  it  disappears  from  the  result  ;  or,  in  algebraic  phrase,  it  is  canceled.    This 
gives  the  equations 

A+B—B=A—B+B=A 

and 

B  —  B  =  Q. 

The  sign  minus  is  called  the  negative  sign  in  ordinary  algebra,  and  any  term 
preceded  by  it  may  be  united  with  it,  and  the  combination  may  be  called  a 
negative  term.  This  use  will  be  adopted  into  all  the  algebras,  with  the  provision 
that  the  derivation  of  the  word  negative  must  not  transmit  its  interpretation. 

22.  The  sign  x  may  be  adopted  from  ordinary  algebra  with  the  name  of 
the  sign  of  multiplication,  but  without  reference  to  the  meaning  of  the  process. 
The  result  of  multiplication  is  to  be  called  the  product.     The  terms  which  are 
combined  by  the  sign  of  multiplication  may  be  called  factors  ;  the  factor  which 
precedes  the  sign  being  distinguished  as  the  multiplier,  and  that  which  follows  it 
being  the  multiplicand.     The  words  multiplier,  multiplicand,  and  product,  may 
also  be  conveniently  replaced  by  the  terms  adopted  by  Hamilton,  of  facient, 
faciend,  andfactum.     Thus  the  equation  of  the  product  is 

multiplier  X  multiplicand  ==  product  ;    or    facient  X  faciend  =  factum. 

When  letters  are  used,  the  sign  of  multiplication  can  be  omitted  as  in  ordinary 
algebra. 


8  PEIRCE  :    Linear  Associative  Algebra. 

23.  When  an  expression  used  as  a  factor  in  certain  combinations  gives  a 
product   which  vanishes,   it  may  be  called  in  those  combinations   a  nil/actor. 
Where  as  the  multiplier  it  produces  vanishing  products  it  is  nilfacient,  but  where 
it  is  the  multiplicand  of  such  a  product  it  is  nilfaciend. 

24.  When   an   expression  used  as  a  factor  in  certain   combinations  over 
powers  the  other  factors  and  is  itself  the  product,  it  may  be  called  an  idem/actor. 
When  in  the  production  of  such  a  result  it  is  the  multiplier,  it  is  idem/orient, 
but  when  it  is  the  multiplicand  it  is  idemfaciend. 

25.  When  an  expression  raised  to  the  square  or  any  higher  power  vanishes, 
it  may  be  called  nilpotent;  but  when,  raised  to  a  square  or  higher  power,  it  gives 
itself  as  the  result,  it  may  be  called  idempotent. 

The  defining  equation  of  nilpotent  and  idempotent  expressions  are  respec 
tively  An  =.  0 ,  and  An  =  A  •  but  with  reference  to  idempotent  expressions,  it 
will  always  be  assumed  that  they  are  of  the  form 

A*=A, 

unless  it  be  otherwise  distinctly  stated. 

26.  Division  is  the  reverse  of  multiplication,  by  which  its  results  are  verified. 
It  is  the  process  for  obtaining  one  of  the  factors  of  a  given  product  when  the 
other  factor  is  given.     It  is  important  to  distinguish  the  position  of  the  given 
factor,  whether  it  is  facient  or  faciend.     This  can  be  readily  indicated  by  com 
bining  the  sign  of  multiplication,  and   placing   it   before    or    after   the    given 
factor  just  as  it  stands  in  the  product.     Thus  when  the  multiplier  is  the  given 
factor,  the  correct  equation  of  division  is 

dividend 
quotient  =    ,— — 

divisor  x 

and  the  equation  of  verification  is 

divisor  X  quotient  =:  dividend. 
But  when  the  multiplicand  is  the  given  factor,  the  equation  of  division  is 

dividend 

quotient  =  — =7— ; — 
X  divisor 

and  the  equation  of  verification  is 

quotient  X  divisor  =  dividend. 

27.  Exponents  may  be  introduced  just  as  in  ordinary  algebra,  and  they 
may  even  be  permitted  to  assume  the  forms  of  the  algebra  under  discussion. 


PEIRCE  :   Linear  Associative  Algebra.  9 

There  seems  to  be  no  necessary  restriction  to  giving  them  even  a  wider  range 
and  introducing  into  one  algebra  the  exponents  from  another.  Other  signs  will 
be  defined  when  they  are  needed. 

The  definition  of  the  fundamental  operations  is  an  essential  part  of  the 
vocabulary,  but  as  it  is  subject  to  the  rules  of  grammar  which  may  be  adopted, 
it  must  be  reserved  for  special  investigation  in  the  different  algebras. 

The   Grammar. 

28.  Quantity    enters    as    a   form    of  thought  into  every  inference.      It  is 
always  implied  in  the  syllogism.     It  may  not,  however,  be  the  direct  object  of 
inquiry ;  so  that  there  may  be  logical  and  chemical  algebras  into  which  it  only 
enters  accidentally,  agreeably  to  §  1.     But  where  it  is  recognized,  it  should  be 
received  in   its   most   general    form    and    in    all   its    variety.     The    algebra   is 
otherwise  unnecessarily  restricted,  and  cannot  enjoy  the  benefit  of   the    most 
fruitful  forms  of  philosophical  discussion.     But  while  it  is  thus  introduced  as  a 
part  of  the  formal  algebra,  it  is  subject  to  every  degree  and  kind  of  limitation  in 
its  interpretation. 

The  free  introduction  of  quantity  into  an  algebra  does  not  even  involve  the 
reception  of  its  unit  as  one  of  the  independent  units  of  the  algebra.  But  it  is 
probable  that  without  such  a  unit,  no  algebra  is  adapted  to  useful  investigation. 
It  is  so  admitted  into  quaternions,  and  its  admission  seems  to  have  misled  some 
philosophers  into  the  opinion  that  quaternions  is  a  triple  and  not  a  quadruple 
algebra.  This  will  be  the  more  evident  from  the  form  in  which  quaternions 
first  present  themselves  in  the  present  investigation,  and  in  which  the  unit  of 
quantity  is  not  distinctly  recognizable  without  a  transmutation  of  the  form.* 

29.  The  introduction  of  quantity  into  an  algebra  naturally  carries  with  it, 
not  only  the  notation  of  ordinary  algebra,  but  likewise  many  of  the  rules  to 
which  it  is  subject.     Thus,  when  a  quantity  is  a  factor  of  a  product,  it  has  the 

*  Hamilton's  total  exclusion  of  the  imaginary  of  ordinary  algebra  from  the  calculus  as  -well  as  from 
the  interpretation  of  quaternions  will  not  probably  be  accepted  in  the  future  development  of  this 
algebra.  It  evinces  the  resources  of  his  genius  that  he  was  able  to  accomplish  his  investigations  under 
these  trammels.  But  like  the  restrictions  of  the  ancient  geometry,  they  are  inconsistent  with  the 
generalizations  and  broad  philosophy  of  modern  science.  With  the  restoration  of  the  ordinary  imaginary, 
quaternions  becomes  Hamilton's  biquaternions.  From  this  point  of  view,  all  the  algebras  of  this  research 
would  be  called  bi-algebras.  But  with  the  ordinary  imaginary  is  involved  a  vast  power  of  research,  and 
the  distinction  of  names  should  correspond  :  and  the  algebra  which  loses  it  should  have  its  restricted 
nature  indicated  by  such  a  name  as  that  of  a  semi-algebra. 


10  PEIECE  :  Linear  Associative  Algebra. 

same  influence  whether  it  be  facient  or  faciend,  so  that  with  the  notation  of 
S  14,  there  is  the  equation 

O  -1  4  A 

Aa  =  a  A , 

and  in  such  a  product,  the  quantity  a  may  be  called  the  coefficient. 

In  like  manner,  terms  which  only  differ  in  their  coefficients,  may  be  added 
by  adding  their  coefficients  ;  thus, 

(a  ±  6)  A  =  a  A  it  bA  —  Aa  =b  Ab  =  A  (a  =b  ft) . 

30.  The  exceeding  simplicity  of  the  conception  of  an  equation  involves  the 

identity  of  the  equations 

A  =  B   and  B  =  A 

and  the  substitution  of  B  for  A  in  every  expression,  so  that 

MA±  C=.MB±  C, 

or  that,  the  members  of  an  equation  may  be  mutually  transposed  or  simultaneously 
increased  or  decreased  or  multiplied  or  divided  by  equal  expressions. 

31.  How  far  the  principle   of   §  16    limits  the   extent  within  which  the 
ordinary  symbols  may  be  used,  cannot  easily  be  decided.     But  it  suggests  limi 
tations  which  may  be  adopted  during  the  present  discussion,  and  leave  an  ample 
field  for  curious  investigation. 

The  distributive  principle  of  multiplication  may  be  adopted ;  namely,  the 
principle  that  the  product  of  an  algebraic  sum  of  factors  into  or  by  a  common 
factor,  is  equal  to  the  corresponding  algebraic  sum  of  the  individual  products  of 
the  various  factors  into  or  by  the  common  factor ;  and  it  is  expressed  by  the 

equations  i(\ 

(A±B)C=A$±  EG. 

C(A±B)=CA±  CB. 

32.  TJie  associative  principle  of  multiplication  may  be  adopted  ;  namely,  that 
the  product  of  successive  multiplications  is  not  affected  by  the  order  in  which  the 
multiplications  are  performed,  provided  there  is  no  change  in  the  relative  position 
of  the  factors ;  and  it  is  expressed  by  the  equations 

ABC=(AB)G=A(BC). 

This  is  quite  an  important  limitation,  and  the  algebras  which  are  subject  to  it 
will  be  called  associative. 


PEIRCE  :    Linear  Associative  Algebra.  11 

33.  The  principle  that  the  value  of  a  product  is  not  affected  by  the  relative 
position  of  the  factors  is  called  the  commutative  principle,  and  is  expressed  by  the 
equation 

AB  —  BA. 

This  principle  is  not  adopted  in  the  present  investigation. 

34.  An  algebra  in  which  every  expression  is  reducible  to  the  form  of  an 
algebraic  sum  of  terms,  each  of  which  consists  of  a  single  letter  with  a  quanti 
tative  coefficient,  is  called  a  linear  algebra*     Such  are  all  the  algebras  of  the 
present  investigation. 

35.  Wherever  there  is  a   limited   number  of  independent    conceptions,  a 
linear  algebra  may  be  adopted.     For  a  combination  which  was  not  reducible  to 
such   an    algebraic   sum    as    those  of  linear  algebra,  would  be  to  that  extent 
independent  of  the  original  conceptions,  and  would  be  an  independent  conception 
additional   to  those    which   were    assumed    to    constitute    the    elements   of  the 
algebra. 

36.  An  algebra  in  which  there  can  be  complete  interchange  of  its  indepen 
dent   units,    without    changing   the    formulae    of    combination,    is    a  completely 
symmetrical  algebra;  and  one  in  which  there  may  be  a  partial  interchange  of  its 
units  is  partially  symmetrical     But  the  term  symmetrical  should  not  be  applied, 
unless  the  interchange  is  more  extensive  than  that  involved  in  the  distributive 
and  commutative  principles.     An  algebra  in  which  the  interchange  is  effected  in 
a  certain  order  which  returns  into  itself  is  a  cyclic  algebra. 

Thus,   quaternions  is  a  cyclic  algebra,  because  in  any  of  its  fundamental 
equations,  such  as 

#  =  —  1 
ij  =  —ji  =  k 
ij~k  =  —  1 

there  can  be  an  interchange  of  the  letters  in  the  order  i ,  j ,  k,  i,  each  letter 
being  changed  into  that  which  follows  it.     The  double  algebra  in  which 


*  In  the  various  algebras  of  De  Morgan's  "  Triple  Algebra,"  the  distributive,  associative  and  com 
mutative  principles  were  all  adopted,  and  they  were  all  linear.  [De  Morgan's  algebras  are  "  semi- 
algebras.1'  See  Cambridge  Phil.  Trans.,  viii.  241.]  [C.  S.  P.] 


! 

is  oyolio  hooauso  the  letters  :nv  intorohangoahlo  in  the  ordor  »'.  ./.  »'.     Hut  neither 
of  those  algobras  is  commutative. 

ST.  \Vhou  an  algebra  ran  bo  roduood  to  a  form  in  whioh  all  the  KM  tors  :\ro 
expressed  :is  powON  of  MUM  ono  of  thorn.  it  may  bo  oallod  a  potential  nbjelmi. 
If  the  po\\ors  ;uv  all  squares.  it  may  bo  railed  ,/mK/r,r/»V  :  if  thoy  aro  oubos.  it 
max  bo  oallod  eiibie  ;  and  siiuilarlv  in  oihor  OMM 

l.ine<ir     1-     .'.r/iiv 


s.  j4//  ^  MMNM 

ij  eb  /o  (i//  // 

For  it  is  ob\ious  that  iu  tho  equation 


eaoh  letter  eau  be  multiplied  by  an  inteirer.  whieh  jrives  the  form 

-f  «/;)  =  fie»^  +  frtf*  -h 


in  whieh  «i.  />  .  <^  anvl  <1  aro  r  The  integers  oan  have  the  ratios  of  any 

four  ival  number,  so  that  by  simple  division  they  oan  be  roihiooil  to  suoh  real 
uumbei-s.  Other  similar  equations  oan  also  be  formed  by  writing  for  ii  and  b.  «>i 
and  />,.  or  for  ,  and  ,/.  ^  and  </t.  or  by  making;  both  these  substitutions  simulta- 
neously.  If  then  the  two  first  of  those  new  equations  are  multiplied  by  J  and 
the  last  b\  1  :  the  sum  of  the  four  equations  will  be  the  same  as  that  whieh 
ild  be  obtained  by  substituting  for  <i  .  />.  <  and  </.  «i  +  J^,.  b  -h  J^.  o  +  J<\ 
ami  ^4-  Jt/v  Hemv  <i  ,  b.  c  and  </  may  be  any  numbers,  real  or  imaginary,  and  in 
gvneral  whatever  mixtures  .1.  />.  (  x  and  /)  may  IN  of  the  original 

units  under  the  form  of  an  algehraio  sum  of  the  letters  «\  /.  If.  ^o..  we  shall 

have 

(A  +  B)(C  +  /))  =  AC  +  KC  -h  AD  -f  Bl 

whioh  is  the  oomp.-         \    rossion  of  the  distributive  prinoiple. 
i  ?  « 


For  if  .-t  =  i  (<ii)  =  <ii  -f  rtj  -h  <»t^  ^  A 

fi=  v^>  =  W^ftJ  +  \fr+  A, 
^  =  n"  -h  r^y  -h  °Jt  -f  A 


ri-:nu'K:     Linear    Axxoeiatirt     A/</>!>ni. 
il  is  obvious  that  AB  =  ^.al^ij] 


(AB)C=  £  (a(>,e,ijk)  =  A(BC)  =  ABC 

which  is  tin4  general  ex  pression  of  tin'  associative  principle. 

40.  lu  wry  linear  atwocnitirt  alyehra,  there  i*  at  lea*t  one  nl>  inj>ot>  nt  or  one 
•nil  potent  c.ry>/v.vs  /<>//. 

Take  anv  combination  of  letters  at  will  and  denote  it  by  A.  Its  square 
is  generally  independent  of  A,  and  its  cube  may  also  be  independent  of  A 
and  J8.  Hut  the  number  of  powers  of  A  that  are  independent  of  A  and  of 
each  other,  cannot  exceed  the  number  of  letters  of  the  alphabet  ;  so  that  there 
must  be  some  least  power  of  A  which  is  dependent  upon  the  interior  powers. 
The  mutual  dependence  of  the  powers  of  A  may  be  expressed  in  the  form  of  an 
equation  of  which  the  tirst  member  is  an  algebraic  sum.  such  as 

V     (n      Am\  —  () 
-m\(t'm'a'    )  —  U> 

All  the  terms  of  this  equation  that,  involve  the  square  and  higher  powers  of  A 
may  be  combined  and  expressed  as  />.!,  so  that  II  is  itself  an  algebraic  sum  of 
powers  of  A,  and  the  equation  may  be  written 

BA  +  «lA  =  (B  +  al)A  =  0. 

It  is  easy  to  deduce  from  this  equal  ion  successively 

B  +  <iAn=       0 


"l/  «! 

so  that  is  an   idempotent    expression.      Hut    if  al  vanishes,   this  expression 

becomes  infinite,  and  instead  of  it  we  have  the  equation 

so  that  B  is  a  m'lpotent  expression. 

41.  When  there  is  an  icU'M/wfent  expression  in  a  linear  associative  algebra,  it 

can  be  assumed  as  one  of  the  independent    units,   and   be   represented  by  one  of 
f/it  letter*  of  tin  alfthalxt  ;  and  it  may  be  called  the  httsi*. 

The  nniaininy  unit*  ean  i><  *<>  .«.<  l<ctnl  a*  to  />t  M  />arat>h  into  four  dixftnet  (jroni>*. 

With  /•<  ft  /•>  ne-i  to  tJi<  l><i*i*,  f/it  unfa  of  the  first  group  an  !•  l<  mfat'fnr* ;  fh<>*<  of 

the  MCOIH/  <//•<>///>  <in   idemfaeicnd' and  ni/faeienf ;   those  of  Me  third  <jroii)>  are   id<  in- 

farirnf  <(/K/  ni/fan't  iul :   ami  fho*e  of  /At  fourth  </roi<fi  an   /n'/farf<>r«. 


1  4  PEIRCE  :    Linear  Associative  Algebra. 

First.  The  possibility  of  the  selection  of  all  the  remaining  units  as  idem- 
faciend  or  nilfaciend  is  easily  established.  For  if  i  is  the  idempotent  base,  its 
definition  gives 

$  =•  i  .  , 

The  product  by  the  basis  of  another  expression  such  as  A   may  be  represented 

• 

by  B,  so  that 

iA  =  B, 

which  gives 

iJB  =  i*A  =  iA  =  B 


whence  it  appears  that  B  is  idemfaciend  and  A  —  B  is  nilfaciend.  In  other 
words,  A  is  divided  into  two  parts,  of  which  one  is  idemfaciend  and  the  other  is 
nilfaciend  ;  but  either  of  these  parts  may  be  wanting,  so  as  to  leave  A  wholly 
idemfaciend  or  wholly  nilfaciend. 

Secondly.  The  still  farther  subdivision  of  these  portions  into  idemfacient  and 
nilfacient  is  easily  shown  to  be  possible  by  this  same  method,  with  the  mere 
reversal  of  the  relative  position  of  the  factors.  Hence  are  obtained  the  required 
four  groups. 

The  basis  itself  may  be  regarded  as  belonging  to  the  first  group. 

42.  Any  algebraic  sum  of  the  letters  of  a  group  is  an  expression  which 
belongs  to  the  same  group,  and  may  be  called  factorially  homogeneous. 

43.  The  product  of  two  factorially  homogeneous   expressions,    which   does   not 
vanish,  is  itself  factorially  homogeneous,  and  its  faciend  name  is  the  same  as  that 
of  its  facient,  while  its  facient  name  is  tJie  same  as  that  of  its  faciend. 

Thus,  if  A  and  B  are,  each  of  them,  factorially  homogeneous,  they  satisfy 
the  equations 

i(AB)  =  (iA)B  , 
(AB)i  = 


which  shows  that  the  nature  of  the  product  as  a  faciend  is  the  same  as  that  of 
the  facient  A,  and  its  nature  as  a  facient  is  the  same  as  that  of  the  faciend  B. 
44.  Hence,  no  product  which  does  not  vanish  can  be  commutative  unless  both  its 
factors  btlong  to  tlw  same  group. 


PEIRCE  :    Linear  Associative  Algebra. 


15 


45.  Every  product  vanishes,  of  which  the  facient  is  idemfacient  while  thefaciend 
is  nilfaciend  ;  or  of  which  the  facient  is  nilfacient  while  the  faciend  is  idemfaciend. 
For  in  either  case  this  product  involves  the  equation 


46.  The  combination  of  the  propositions  of  §§  43  and  45  is  expressed  in  the 
following  form  of  a  multiplication  table.  In  this  table,  each  factor  is  expressed 
by  two  letters,  of  which  the  first  denotes  its  name  as  a  faciend  and  the  second  as 
a  facient.  The  two  letters  are  d  and  n,  of  which  d  stands  for  idem  and  n  for  nil. 
The  facient  is  written  in  the  left  hand  column  of  the  table  and  the  faciend  in  the 
upper  line.  The  character  of  the  product,  when  it  does  not  vanish,  is  denoted 
by  the  combination  of  letters,  or  when  it  must  vanish,  by  the  zero,  which  is 
written  upon  the  same  line  with  the  facient  and  in  a  column  under  the  faciend. 

dd       dn      nd 


nn 


dd 


dn 


nd 


nn 


dd 

dn 

0 

0 

0 

0 

dd 

dn 

nd 

nn 

0 

0 

0 

0 

nd 

nn 

i 

47.  It  is  apparent  from  the  inspection  of  this  table,   that  every  expression 
which  belongs  to  the  second  or  third  group  is  nilpotent. 

48.  It  is  apparent  that  all  commutative  products  which   do   not   vanish   are 
restricted  to  the  first  and  fourth  groups. 

49.  It  is  apparent  that  every  continuous  product  which  does  not  vanish,  has 
the  same  faciend  name  as  its  first  facient,  and  the  same  facient  name  as  its  last 
faciend. 

50.  Since  the  products  of  the  units  of  a  group  remain  in  the  group,  they 
cannot  serve  as  the  bond  for  uniting  different  groups,  which  are  the  necessary 
conditions  of  a  pure  algebra.     Neither  can  the  first  and  fourth  groups  be  con 
nected  by  direct  multiplication,  because  the  products  vanish.     The  first  and  fourth 
groups,  therefore,  require  for  their  indissoluble  union  into  a  pure  algebra  that  there 
should  be  units  in  each  of  the  other  two  groups. 


16  PEIRCE  :    Linear  Associative  Algebra. 

51.  In  an  algebra  which  has  more  than  two  independent  units,  it  cannot 
happen  that  all  the  units  except  the  base  belong  to  the  second  or  to  the  third  group. 
For  in  this  case,  each  unit  taken  with  the  base  would  constitute  a  double  algebra, 
and  there  could  be   no  bond  of  connection  to   prevent  their  separation  into 
distinct  algebras. 

52.  The  units  of  the  fourth  group  are  subject  to  independent  discussion,  as  if  they 
constituted  an  algebra  of  themselves.     There  must  be  in  this  group  an  idempotent 
or  a  nilpotent  unit.     If  there  is  an  idempotent  unit,  it  can  be  adopted  as  the 
basis  of  this  group,  through  which  the  group   can   be  subdivided  into  subsidiary 
groups. 

The  idempotent  unit  of  the  fourth  group  can  even  be  made  the  basis  of  the 
whole  algebra,  and  the  first,  second  and  third  groups  will  respectively  become 
the  fourth,  third  and  second  groups  for  the  new  basis. 

53.  When  the  first  group  comprises  any  units  except  the  basis,  there  is  beside* 
the   basis  another  idempotent  expression,  or  there  is  a  nilpotent  expression.     By  a 
process  similar  to  that  of  §  40  and  a  similar  argument,  it  may  be  shown  that  for 
any  expression  A,  which  belongs  to  the  first  group,  there  is  some  least  power 
which  can  be  expressed  by  means  of  the  basis  and  the  inferior  powers  in  the 
form  of  an  algebraic  sum.     This  condition  may  be  expressed  by  the  equation 


If  then  h  is  determined  by  the  ordinary  algebraic  equation 

s.  («**-)+&=<>, 

and  if 

Al  =  A  —  hi 

is  substituted  for  A  ,  an  equation  is  obtained  between  the  powers  of  A  ,  from 
which  an  idempotent  expression,  B,  or  else  a  nilpotent  expression,  can  be 
deduced  precisely  as  in  §  40.* 

54.  Wlien  there  is  a  second  idempotent  unit  in  the  first  group,  the  basis  can  be 
changed  so  as  to  free  the  first  group  from  this  second  idempotent  unit. 

Thus  if  i  is  the  basis,  and  if  j  is  the  second  idempotent  unit  of  the  first 
group,  the  basis  can  be  changed  to 

*  The  equation  in  h  may  have  no  algebraic  solution,  in  which  case  the  new  idempotent  or  nilpotent 
would  not  be  a  direct  algebraic  function  of  i  and  A.     [C.  S.  P.] 


PEIRCE  :   Linear  Associative  Algebra.  17 


and  with  this  new  basis,  j  passes  from  the  first  to  the  fourth  group.     For 
First,  the  new  basis  is  idempotent,  since 

*!  =  (*  —  y)2  =  **  —  2i?  +ys  =  i  —  y  =  i'i  ; 
and  secondly,  the  idempotent  unit  y  passes  into  the  fourth  group,  since 

hj=(i—j)j=  *y—  y8=y—  y  =  o  , 
y%  =y  (*  —  y)  =y*  —  y2  =y  —  y  =  o  . 

55.  TF^A  £Ae  preceding  change  of  basis,  expressions  may  pass  from  idemfacient 
to  nilfacient,  or  from  idemfaciend  to  nilfaciend,  but  not  the  reverse. 

For  first,  if  A  is  nilfacient  with  reference  to  the  original  basis,  it  is  also,  by 
§  45,  nilfacient  with  reference  to  the  new  basis  ;  or  if  it  is  nilfaciend  with 
reference  to  the  original  basis,  it  is  nilfaciend  with  reference  to  the  new  basis. 

Secondly,  all  expressions  which  are  idemfacient  with  reference  to  the 
original  basis,  can,  by  the  process  of  §  41,  be  separated  into  two  portions  with 
reference  to  the  new  basis,  of  which  portions  one  is  idemfacient  and  the 
other  is  nilfacient  ;  so  that  the  idemfacient  portion  remains  idemfacient,  and  the 
remainder  passes  from  being  idemfacient  to  being  nilfacient.  The  same  process 
may  be  applied  to  the  faciends  with  similar  conclusions. 

56.  It  is  evident,  then,  that  each  group*  can  be  reduced  so  as  not  to  contain 
more  than  one  idempotent  unit,  which  will  be  its  basis.     In  the  groups  which 
bear  to  the   basis  the   relations  of  second  and  third  groups,   there   are   only 
nilpotent  expressions. 

57.  In  a  group  or   an  algebra  which   has  no  idempotent  expression,   all  the 
expressions  are  nilpotent. 

Take  any  expression  of  this  group  or  algebra  and  denote  it  by  A.  If  no 
power  of  A  vanished,  there  must  be,  as  shown  in  §  40,  some  equation  between 

the  powers  of  A  of  the  form 

2mamAm=0, 

in  which  al  must  vanish,  or  else  there  would  be  an  idempotent  expression  as  is 
shown  in  §  40,  which  is  contrary  to  the  present  hypothesis.     If  then  m0  denote 

*  That  is,  the  first  group  as  well  as  each  of  the  subsidiary  groups  of  \  52.     [C.  S.  P.] 


18  PEIRCE  :    Linear  Associative  Algebra. 

the  exponent  of  the  least  power  of  A  that  entered  into  this  equation,  and  m0  +  h 
the  exponent  of  the  highest  power  that  occurred  in  it,  the  whole  number  of 
terms  of  the  equation  would  be,  at  most,  h  +  1  .  If,  now,  the  equation  were 
multiplied  successively  by  A  and  by  each  of  its  powers  as  high  as  that  of  which 
the  exponent  is  (w0—  1)A,  this  highest  exponent  would  denote  the  number  of 
new  equations  which  would  be  thus  obtained.  If,  moreover, 

B  =  Am°  , 
then  the  highest  power  of  A  introduced  into  these  equations  would  be 


The  whole  number  of  powers  of  A  contained  in  the  equations  would  be  mji  -f  1  , 
and  7i  +  1  of  these  would  always  be  integral  powers  of  B  ;  and  there  would 
remain  (m0  —  l)h  in  number  which  were  not  integral  powers  of  B.  There 
would  be,  therefore,  equations  enough  to  eliminate  all  the  powers  of  A  that 
were  not  integral  powers  of  B  and  still  leave  an  equation  between  the  integral 
powers  of  B  •  and  this  would  generally  include  the  first  power  of  B.  From 
this  equation,  an  idempotent  expression  could  be  obtained  by  the  process  of  §  40, 
which  is  contrary  to  the  hypothesis  of  the  proposition. 

Therefore  it  cannot  be  the  case  that  there  is  any  equation  such  as  that  here 
assumed  ;  and  therefore  there  can  be  no  expression  which  is  not  nilpotent.  The 
few  cases  of  peculiar  doubt  can  readily  be  solved  as  they  occur;  but  they 
always  must  involve  the  possibility  of  an  equation  between  fewer  powers  of  B 
than  those  in  the  equation  in  A* 

58.  When  an  expression  is  nilpotent,  all  its  powers  which  do  not  vanish  are 
mutually  independent. 

Let  A  be  the  nilpotent  expression,  of  which  the  ?ith  power  is  the  highest 
which  does  not  vanish.  There  cannot  be  any  equation  between  these  powers 
of  the  form 


*  In  saying  that  the  equation  in  B  will  generally  include  the  first  power  of  B ,  he  intends  to  waive 
the  question  of  whether  this  always  happens.  For,  he  reasons,  if  this  is  not  the  case  then  the  equation 
in  B  is  to  be  treated  just  as  the  equation  in  A  has  been  treated,  and  such  repetitions  of  the  process  must 
ultimately  produce  an  equation  from  which  either  an  idempotent  expression  could  be  found,  or  else  A 
would  be  proved  nilpotent.  [C.  S.  P.] 


PEIRCB  :    Linear  Associative  Algebra.  19 

For  if  w0  were  the  exponent  of  the  lowest  power  of  A  in  this  equation,  the 
multiplication  of  the  equation  by  the  (n  —  w0)th  power  of  A  reduces  it  to 

am  An  =  0  ,     am  =  0  , 

n  0 

that  is,  the  w0th  power  of  A  disappears  from  the  equation,  or  there  is  no  least 
power  of  A  in  the  equation,  or,  more  definitely,  there  is  no  such  equation. 

59.  In  a  group  or  an  algebra  which  contains  no  idempotent  expression,  any 
expression   may    be   selected   as    the    basis;    lid   one   is  preferable   which   has  the 
greatest  number  of  powers  which  do  not  vanish.     All  the  powers  of  the  basis  which 
do  not  vanish  may  be  adopted  as  independent  units  and  represented  by  the 
letters  of  the  alphabet. 

A  nilpotent  group  or  algebra  may  be  said  to  be  of  the  same  order  with  the  number 
of  powers  of  its  basis  that  do  not  vanish,  provided  the  basis  is  selected  by  the 
preceding  principle.  Thus,  if  the  squares  of  all  its  expressions  vanish,  it  is  of 
the  first  order  ;  if  the  cubes  all  vanish  and  not  all  the  squares,  it  is  of  the  second 
order,  and  so  on. 

60.  It  is  obvious  that  in  a  nilpotent   group  whose  order  equals   the   number 
of  letters    which   it   contains,    all   the  letters  except  the  basis  may  be  taken  as  the 
successive  powers  of  the  basis. 

61.  In   a  nilpotent    group,  every  expression,  such  as  A,   has  some   least 
power  that  is  nilfacient  with  reference  to  any  other  expression,  such  as  B ,  and 
which  corresponds  to  what   may  be    called  the  facient  order  of  B  relatively  to 
A  ;  and  in  the  same  way,  there  is  some  least  power  of  A  which  is  nilfaciend  with 
reference  to  B  ,  and  which  corresponds  to  the  faciend  order  of  B  relatively  to  A. 
When  the  facient  and  faciend  orders  are  treated  of  irrespective  of  any  especial 
reference,  they  must  be  referred  to  the  base. 

The  facient  order  of  a  product  which  does  not  vanish,  is  not  higher  than  that  of 
its  facient ;  and  the  faciend  order  is  not  higher  than  that  of  its  faciend. 

62.  After  the  selection  of  the  basis  of  a  nilpotent  group,  some  one  from 
among  the  expressions  which  are  independent  of  the  basis  may  be  selected  by  the 
same  method  by  which  the  basis  was  itself  selected,  which,  together  with  all  its 
powers  that  are  independent  of  tlie  basis,  may  be  adopted  as  new  letters  ;  and  again, 
from  the  independent  expressions  which  remain,  new  letters  may  be  selected  by  the 
same  process,  and  so  on  until  the  alphabet  is  completed.    In  making  these  selections, 
regard  should  be  had  to  the  factorial  orders  of  the  products. 


20  PEIRCE  :    Linear  Associative  Algebra. 

63.  In  every  nilpotent  group,  the  facient  order  of  any  letter  which  is  indepen 
dent  of  the  basis  can  be  assumed  to  be  as  low  as  the  number  of  letters  which  are 
independent  of  the  basis. 

Thus,  if  the  number  of  letters  which  are  independent  of  the  basis  is  denoted 
by  n' ,  and  if  n  is  the  order  of  the  group  (and  for  the  present  purpose  it  is  suffi 
cient  to  regard  n'  as  being  less  than  n),  it  is  evident  that  any  expression,  A ,  with 
its  successive  products  by  the  powers  of  the  basis  i ,  as  high  as  the  w'th,  and  the 
powers  of  the  basis  which  do  not  vanish,  cannot  all  be  independent  of  one 
another ;  so  that  there  must  be  an  equation  of  the  form 

n  n' 

I  om 

Accordingly,  it  is  easy  to  see  that  there  is  always  a  value  of  Al  of  the  form 


which  will  give 

imAt  =  0 , 

which  corresponds  to  the  condition  of  this  section. 

There  is  a  similar  condition  which  holds  in  every  selection  of  a  new  letter  by  the 
method  of  the  preceding  section. 

64.  In  a  nilpotent  group,  the  order  of  which  is  less  by  unity  than  the  number  of 
letters,  the  letter  which  is  independent  of  the  basis  and  its  powers  may  be  so  selected 
that  its  product  into  the  basis  shall  be  equal  to  the  highest  power  of  tlw  basis  which 
does  not  vanish,  and  that  its  square  shall  either  vanish  or  shall  also  be  equal  to  the 
highest  power  of  the  basis  that  does  not  vanish.  Thus,  if  the  basis  is  i ,  and  if  the 
order  of  the  algebra  is  n ,  and  if  j  is  the  remaining  letter,  it  is  obvious,  from  §  63, 
that/  might  have  been  assumed  such  that 

#=r&, 

which  gives 

iji  =  ij*  —  0  ; 

and  therefore,  ji  =  ain  +  bj , 

j*  =  a'in  +  bj, 
0  =  jin  + x  —  bjin  —  bnji  —  b , 
ji  —  ain, 

j*i  =  ajP  =  0  =  b'j*  =  //, 
/2  —  a'in  • 


PEIRCE  :    Linear  Associative  Algebra.  21 

so  that  if  7  _ 


we  have 

j\il=  i?=.j\, 

and  it  and/!  can  be  substituted  for  i  and/,  which  conforms  to  the  proposition 
enunciated. 

It  must  be  observed,  however,  that  the  analysis  needs  correction  when  the 
group  is  of  the  second  order. 

65.  In  a  nilpotent  group  of  the  first  order,  the  sign   of  a  product  is  merely 
reversed  by  changing  the  order  of  its  factors.     Thus,  if 

A*  =  B*  =  (A  +  B)2  =  0 
it  follows  by  development,  that 

(A  -\-  B}2  =  Az  -j-  AB  -\-  BA  -f-  B?  =•  AB  -f-  BA  =  0 
BA  =  —  AB, 

which  is  the  proposition  enunciated. 

66.  In  general,  in  any  nilpotent  group  of  the  ?ith  order,  if  (A8,   B*)  denotes  the 
sum  of  all  possible  products  of  the  form 

ApBq  Ap'Bq'  Ap"Bq"  .  .  . 
in  which 

2p  =  * ,     2q  =  t, 

and  if 

i    ,  ___       i    -i 

it  will  be  found  that 

For  since  (A  +  xB)n  + 1  =  0 

whatever  be  the  value  of  x ,  the  multiplier  of  each  power  of  x  must  vanish,  which 

gives  the  proposed  equation 

(A8,  Bt)=Q. 

67.  In  the  first  group  of  an  algebra,  having  an  idempotent  basis,  all  the  expres 
sions  except  the  basis  may  be  assumed  to  be  nilpotent.     For,  by  the  same  argument 
as  that  of  §53,  any  equation  between  an  expression  and  its  successive  powers 
and  the  basis  must  involve  an  equation  between  another  expression  which  is 


22  PEIRCE  :    Linear  Associative  Algebra. 

easily  defined  and  its  successive  powers  without  including  the  basis.  But  it 
follows  from  the  argument  of  §57,  that  such  an  equation  indicates  a  corres 
ponding  idempotent  expression  ;  whereas  it  is  here  assumed  that,  in  accordance 
with  §56,  each  group  has  been  brought  to  a  form  which  does  not  contain  any 
other  idempotent  expression  than  the  basis.  It  must  be,  therefore,  that  all  the 
other  expressions  are  nilpotent. 

68.  No  product  of  expressions  in  the  first  group  of  an  algebra  having  an  idem- 
potent  basis,  contains  a  term  which  is  a  multiple  of  the  basis. 

For,  assume  the  equation 

AB-—  xi+  C, 

in  which  A,  B  and  C  are  nilpotents  of  the  orders  m  ,  n  and  p,  respectively. 

Then, 

=  —  xAm  +  AmC 


that  is,  the  term  —  xi  vanishes  from  the  product  AB. 

69.  It  follows,  from  the  preceding  section,  that  if  the  idempotent  basis  were 
taken  away  from  the  first  group  of  which  it  is  the  basis,  the  remaining  letters  of  the 
first  group  would  constitute  by  tliemselves  a  nilpotent  algebra. 

Conversely,  any  nilpotent  algebra  may  be  converted  into  an  algebra  with  an 
idempotent  basis,  by  the  simple  annexation  of  a  letter  idemfaciend  and  idemfacient 
with  reference  to  every  other* 

70.  However    incapable    of   interpretation    the  nilfactorial    and    nilpotent 
expressions  may  appear,  they  are  obviously  an  essential  element  of  the  calculus 
of  linear  algebras.     Unwillingness  to  accept  them  has  retarded  the  progress  of 
discovery  and  the  investigation  of  quantitative  algebras.     But  the  idempotent 
basis  seems  to  be  equally  essential  to  actual  interpretation.     The  purely  nilpotent 
algebra  may  therefore  be  regarded  as  an  ideal  abstraction,  which  requires  the 
introduction  of  an  idempotent  basis,  to  give  it  any  position  in  the  real  universe. 
In  the  subsequent  investigations,  therefore,  the  purely  nilpotent  algebras  must 
be  regarded  as  the  first  steps  towards  the  discovery  of  algebras  of  a  higher 
degree  resting  upon  an  idempotent  basis. 

*  That  every  such  algebra  must  be  a  pure  one  is  plain,  because  the  algebra  (aa)  is  so.     [C.  S.  P.] 


PEIRCB  :    Linear  Associative  Algebra.  23 

71.  Sufficient  preparation  is  now  made  for  the 

INVESTIGATION   OF   SPECIAL  ALGEBRAS. 

The  following  notation  will  be  adopted  in  these  researches.  Conformably  with 
§9,  the  letters  of  the  alphabet  will  be  denoted  by  i,  j  ,  k,  I,  m  and  n.  To 
these  letters  will  also  be  respectively  assigned  the  numbers  1  ,  2  ,  3  ,  4  ,  5  and 
6.  Moreover,  their  coefficients  in  an  algebraic  sum  will  be  denoted  by  the 
letters  a,  6,  c,  d,  e  and/.  Thus,  the  product  of  any  two  letters  will  be 
expressed  by  an  algebraic  sum,  and  below  each  coefficient  will  be  written  in 
order  the  numbers  which  are  appropriate  to  the  factors.  Thus, 

jl  =  au  i  +  624  j  +  C24  &  +  ^24  Z  +  e24  m  +  /24  n  , 
while 

lj  =  a&i  +  &42  j  H-  c42  k  +  c?42  I  +  e42 


In  the  case  of  a  square,  only  one  number  need  be  written  below  the  coefficient, 

thus 

tf  =a3i  +  bsj  +  c3k  +  d3l  +  e3m+f3n. 

The  investigation  simply  consists  in  the  determination  of  the  values  of  the 
coefficients,  corresponding  to  every  variety  of  linear  algebra  ;  and  the  resulting 
products  can  be  arranged  in  a  tabular  form  which  may  be  called  the  multipli 
cation-table  of  the  algebra.  Upon  this  table  rests  all  the  peculiarity  of  the 
calculus.  In  each  of  the  algebras,  it  admits  of  many  transformations,  and  much 
corresponding  speculation.  The  basis  will  be  denoted  by  i  . 

72.  The  distinguishing  of  the  successive  cases  by  the  introduction  of 
numbers  will  explain  itself,  and  is  an  indispensable  protection  from  omission 
of  important  steps  in  the  discussion. 

SINGLE  ALGEBRA. 

Since  in  a  single  algebra  there  is  only  one  independent  unit,  it  requires  no 
distinguishing  letter.  It  is  also  obvious  that  there  can  be  no  -single  algebra 
which  is  not  associative  and  commutative.  Single  algebra  has,  however,  two 

cases  : 

[1],  when  its  unit  is  idempotent  ; 

[2],  when  it  is  nilpotent. 
[1].  The  defining  equation  of  this  case  is 


24  PEIRCE  :    Linear  Associative  Algebra. 

This  algebra  may  be  called  (a-^  and  its  multiplication  table  is  * 


[2].  The  defining  equation  of  this  case  is 

*2  =  0 . 
This  algebra  may  be  called  (^)  and  its  multiplication  table  is  f 


DOUBLE  ALGEBRA. 
There  are  two  cases  of  double  algebra : 

[1],  when  it  has  an  idernpotent  expression ; 
[2],  when  it  is  nilpotent. 

[1].  The  defining  equation  of  this  case  is 

By  §§41  and  50,  there  are  two  cases : 

[I2],  when  the  other  unit  belongs  to  the  first  group  ; 
[12],  when  it  is  of  the  second  group. 

The    hypothesis    that   the    other   unit  belongs  to  the  third  group  is  a  virtual 
repetition  of  [12]. 

[I2].  The  defining  equations  of  this  case  are 

It  follows  from  §§  67  and  69,  that  there  is  a  double  algebra  derived  from  (JJ 
which  may  be  called  («2) ,  of  which  the  multiplication  table  is  J 

*  This  algebra  may  be  represented  by  i  —  A  :  A  in  the  logic  of  relatives.     See  Addenda.     [C.  S.  P.] 

t  This  algebra  takes  the  form  i  —  A  :  B ,  in  the  logic  of  relatives.     [C.  S.  P.] 

I  This  algebra  may  be  put  in  the  form  i  =  A  :  A  +  B  :  B ,  j  —  A  :  B  .    [C.  S.  P.] 


PEIRCE  :   Linear  Associative  Algebra. 


25 


[12].  The  defining  equations  of  this  case  are,  by  §  41, 

ij  =j ,  ji  =  0 ; 
whence,  by  §  46, 

/*=0. 

A  double  algebra  is  thus  formed,  which  may  be  called  (bz),  of  which  the  multi 
plication  table  is  * 

*         j 


[2].  The  defining  equation  of  this  case  is 

in=  0, 
in  wliich  n  is  the  least  power  of  i  which  vanishes.     There  are  two  cases : 

[21],  when  n  —  3  ; 
[2s],   when  n  =2. 

[21].  The  defining  equation  of  this  case  is 

?=0, 

and  by  §  60, 

*=/- 

Tliis  gives  a  double  algebra  which  may  be  called  (V2),  its  multiplication 
table  being  f 


*  This  algebra  may  be  put  in  the  form  i  =  A  :  A  ,  j—.  A  :  B  .    [C.  S.  P. ] 
t  In  relative  form,  i  —  A:B+B:C.  j  —  A\C,     [C.  S.  P.] 


26 


PEIRCE  :    Linear  Associative  Algebra. 


[22].  The  defining  equations  of  this  case  are 


and  it  follows  from  §§64  and  65  that 


so  that  there  is  no  pure  algebra  in  this  case. 


TRIPLE  ALGEBRA. 


There  are  two  cases : 


[1] ,  when  there  is  an  idempotent  basis ; 
[2],  when  the  basis  is  nilpotent. 

[1].  The  defining  equation  of  this  case  is 


* 


There  are,  by  §§  41,  50  and  51,  three  cases  : 

[I2],   when./  and  /,:  are  both  in  the  first  group  ; 

[12],  whenj"  is  in  the  first,  and  7^  in  the  second  group  ; 

[13],  when./  is  in  the  second,  and  Je  in  the  third  group. 

The   case  of  j  being  in  the  first,  and  &  in  the  third  group,  is  a   virtual 
repetition  of  [12]. 

[I2].  The  defining  equations  of  this  case  are 


*This  case  takes  the  form  i  —  A  :  B,  3  —  C  :  D.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 


27 


It  follows  from  §§  67  and  69,  that  the  only  algebra  of  this  case  may  be  derived 
from  (c8) ;  it  may  be  called  (a3),  and  its  multiplication  table  is  * 


0         0 


[12].  The  defining  equations  of  this  case  are 

ji  —  ij  —  j  ,     ik  =  7f.  ,     Id  =  0  ; 
whence,  by  §§  46  and  67, 

y*  =  /^  =  Zy  =  0,    J7s  =  cuk, 
j*7c  —  0  =  c23t/&  =  c|3  fc  =  c23  =// 

and  there  is  no  pure  algebra  in  this  case.f 

[13].  The  defining  equations  of  this  case  are 

ij  =  j,     Id  =  7c  ,    ji  —  i7c  =•  0  ; 
whence,  by  §  46, 

j9  =  /.:2  =  7ej  =  0  ,     jk  =  a.23  i  , 

jig  =  o  =  aaj  =  «23  =y/i-, 


and  there  is  no  pure  algebra  in  this  case.J 

[2].  The  defining  equation  of  this  case  is 


in  which  n  is  the  lowest  power  of  i  that  vanishes. 

There  are  three  cases  : 

[21],  when  n  =  4  ; 

[21],  when  n—  3; 
[23],  when  w  =  2. 


*In  relative  form,  i  —  A  :A+B:B-\-  C  :  (7,  j=  A  :B  +  B:C,  Tc  —  A:C.    [C.  S.  P.] 
t  That  is  to  say,  i  and  j  by  themselves  form  the  algebra  a2  ,  and  i  and  k  by  themselves  constitute  the 
algebra  &2  ,  while  the  products  of  j  and  k  vanish.     Thus,  the  three  letters  are  not  indissolubly  bound 
together  into  one   algebra.      In  relative  form,  this  case  is,  i  —  A:A-\-B:B,j—A:B,  k  —  A:C. 

[C.  S.  P.] 

I  In  relative  form,  i=A:A+D:D,j=.A:B,*=C:D.    [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 
[21].  The  defining  equation  of  this  case  is 

and  by  §  60 


i4  =  0  , 


This  gives  a  triple  algebra  which  may  be  called  (b3),  the  multiplication  table 
being  * 


Jc 


0 


0 


0 


0 


[22].  The  defining  equation  of  this  case  is 

i3  =  0 , 
and  by  §§  59  and  64,  observing  the  exception, 

?      ^*-       ^  o  /"  *          (I 

There  is  no  pure  algebra  when  b31  vanishes,*)-  and  there  are  two 

[221],  when  b3  does  not  vanish ; 
[23],  when  b3  vanishes. 

[221].  The  defining  equation  of  this  case  can,  without  loss  of  generality, 
be  reduced  to 


This  gives  a  triple  algebra  which  may  be  called  (c3),  the  multiplication  table 
being  J 


cases  : 


Inrelativeform,i  =  A:B  +  B:C+C:D,j  =  A:C+B:D,  Je  =  A:D.     [C.  S  P] 
This  case  takes  the  relative  form,  i=A:B  +  B:C,  j  =  A:C,  Jc  =  b3A  :  D  +  D  :C.     1C  S  P  1 
lIv™tetivefoi-m,i  =  A:B+B:C,j  =  A:C,lc  =  a.A:B  +  A:D  +  D-C.     [C  S  P  ]   ' 


PEIRCE  :   Linear  Associative  Algebra. 
(c3)     i          j         Jc 


29 


j 

0 

0 

0 

0 

0 

aj 

0 

j 

An  interesting  special  example  of  this  case  is  afforded  by  a  .  =  —  2  ,  when 


(k  +  i)  i  =      j 
(*±'?=      0, 

so  that  7j  +  i  might  be  substituted  for  It,  and  in  this  form,  the  multiplication 
table  of  this  algebra,  which  may  be  called  (c'3),  is  * 


j 

0 

j 

0 

0 

0 

—j 

0 

0 

*  In  relative  form,  i  =  A:B  +  B:C,  j—A:C,  k  =  —A  :B+B:C+A  :D+D:C. 
When  a  =  +  2  ,  the  algebra  equally  takes  the  form  (c'3 ) ,  on  substituting  k  —  i  for  k.     On  the  other 
hand,  provided  a  is  neither  2  nor  —  2 ,  the  algebra  may  be  put  in  the  form 


To  effect  the  transformation,  we  write  a  —  —  6  —    and  substitute  i  +  bk  and  i  +  ^  k  for  i  and  &,  and 
V  ~  6/J  for<?"'    Thus  the  alSebra  (ca)  has  two  distinct  and  intransmutable  species,  (c's)  and  (c'3').    [C.  S.  P.] 


30 


PEIRCE  :   Linear  Associative  Algebra. 


[23].  The  defining  equation  of  this  case  is 

P  =  0 , 

and   lsl   may  be    reduced  to  unity  without  loss  of  generality,  giving  a  triple 
algebra  which  may  be  called  (cZ3),  the  multiplication  table  being 

i          j          & 


j 

0 

0 

0 

0 

0 

j 

0 

0 

Iii  this  case 


=  0- 


(i-k?  =0, 


so    that  i  —  7c  may  be  substituted  for   i,    and  in  this  form  the  multiplication 
table  is  * 

MO     *          J          * 


0 

0 

0 

0 

0 

0 

.;' 

0 

0 

[23].  The  defining  equations  of  this  case  are 

.-2  —  .'2  —  13,  —  n 

I     J       Iv     U  , 

and  by  the  principles  of  §§  63  and  65,  it  may  be  assumed  that 

ij  =  — ji  •=•  —  ilf,  =•  Id  =•  0 , 


*In  relative  form,  i  —  B\C,  j  —  A:C,   k  =  A:B.      This  is  the  algebra  of  alio-relations  in  its 
typical  form.     [C.  S.  P.] 


PEIRCE  :   Linear  Associative  Algebra. 


31 


We  thus  get  a  triple  algebra  which  may  be  called  (e3),  its  multiplication  table 
being* 

M     i          J          & 


0 

0 

0 

0 

0 

i 

0 

—  i 

0 

QUADRUPLE  ALGEBRA. 
There  are  two  cases  : 

[1],  when  there  is  an  idempotent  basis ; 
[2],  when  the  base  is  nilpotent. 

[1].  The  defining  equation  of  this  case  is 

•2  • 

There  are  six  cases  : 

[I2],   when/,  &,  and  I,  are  all  in  the  first  group ; 

[12],  when/  and  It,  are  in  the  first,  and  I  in  the  second  group  ; 

[13],  when/  is  in  the  first,  and  It  and  I  in  the  second  group  ; 

[14],  when/  is  in  the  first,  It,  in  the  second,  and  I  in  the  third  group ; 

[15],  when/  and  le  are  in  the  second,  and  I  in  the  third  group  ; 

[16],  when /is  in  the  second,  It,  in  the  third,  and  I  in  the  fourth  group. 

The  other  cases  are  excluded  by  §§50  and  51,  or  are  obviously  virtual  repeti 
tions  of  those  which  are  given. 

[I2].  The  defining  equations  of  this  case  are 


and  from  §§  60  and  69,  the  algebras  (63),  (c3),  (c/3),  and  (es),  give  quadruple 
algebras  which  may  be  named  respectively  (#4),  (64),  (c4),  and  (W4),  their 
multiplication  tables  being 


*  In  relative  form,  i  =  A  :  D  ,  j=  A  :  B—  C:D. 
numbers.     [C.  S.  P.] 


A:C+B:D.    This  is  the  algebra  of  alternate 


32 


PEIRCE  :   Linear  Associative  Algebra. 


Tf 


i 

j 

It 

i 

j 

k 

1 

0 

If 

0 

0 

1 

0 

0 

0 

(c4)     i         j         Tf          I 


i 

j 

Tf 

1 

j 

Tf 

0 

0 

Tf 

0 

0 

0 

1 

Tf 

0 

0 

i 

j 

If 

1 

j 

Tf 

0 

0 

Tf 

0 

0 

0 

1 

ale 

0 

Tf 

i 

j 

Tf 

j 

j 

0 

0 

0 

Tf 

0 

0 

> 

1 

0 

—  J 

* 

0 

The  special  case  (c'3)  gives  a  corresponding  special  case  of  (Z>4),  which  may 
be  called  (&'4),  of  which  the  multiplication  table  is 


i 

j 

& 

1 

j 

Tf 

0 

Tf 

Tf 

0 

0 

0 

1 

£ 

0 

0 

PEIRCE  :    Linear  Associative  Algebra. 


33 


The  second  form  of  (d3]  gives  a  corresponding  second  form  of  (c4).  of  which 
the  multiplication  table  is 

(c4)     i         j         Jc          I 


I 


i 

j 

7c 

I 

j 

0 

0 

0 

k 

0 

0 

0 

I 

* 

0 

0 

[12].  The  defining  equations  of  this  case  are 

if  =  ji  =.  j,     ik  =  Id  •=  It ,     il  ==.  I,     li  =.  0 , 
and  it  follows  from  §§  67  and  69,  that  (c2)  gives 


and  from  §  46, 
whence 


=  Ik  =  Z2  =  0  ,    jl  = 


kl  = 


and  there  is  no  pure  algebra  in  this  case.* 

[13].  The  defining  equations  of  this  case  are 

ij  =ji  =j ,     Hi'  =  If,     il  =  I,     Id  =  li  =  0 , 
which  give  by  §§  46  and  67 

and  it  may  be  assumed  that 

jk  =•  I,    whence  jl  =  0  . 

This   gives   a    quadruple    algebra  which  may  be  called  (e4),  its  multiplication 
table  being  f 

*In   relative    form,    i  —  A  :  A  +  B  :B  +  C :  C+  D  :  JD,   j  =  A:B  +  B:C,    k  =  A:C,    l=D:C. 
[C.  S.  P.] 

t  In  relative  form,  i=  A  :  A  +  B  :  B ,  j—  A  :  B ,  k  —  B  :  C,  I  =  A  :  C.     [C.  S.  P.] 
VOL.  IV. 


34 


PEIRCE  :    Linear  Associative  Algebra. 
(e4)     i          j          k          I 


k 


I 


i 

.; 

k 

I 

j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[14].  The  defining  equations  of  this  case  are 

if  =  ji  =  j,     ik  =  7c,     li  =  I,     Id  =  il  =  0  ; 
which  give,  by  §§  46  and  67, 

0  =j*  =jl  =  kj  —  7t*  =  lie  =  P, 
jk  =  c^k,     7j  =  d^l,     Id—  aui  + 
0  =jz7c  =  cz3j7t  =  cffl  =  c2S  =j7c, 
0  =  If  =  d^j  =  dU  =  d4Z  =  Ij, 

0  =kl=0  —  « 


and  b3i  cannot  be  permitted  to  vanish,*  so  that  it  does  not  lessen  the  generality 
to  assume 

7d  =  j. 

This  gives  a  quadruple  algebra  which  may  be  called    (/4),    its    multiplication 
table  being  f 


i 

j 

k 

0 

j 

0 

0 

0 

0 

0 

0 

;' 

I 

0 

0 

0 

*  For  then  the  algebra  would  split  up  into  three  double  algebras.     [C.  S.  P.] 
tin  relative  form,  i  =  A  :  A  + B  :B ,  j=  A  :B ,  k  =  A:C,  l~C:B.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra.  35 


[15].  The  defining  equations  of  this  case  are 

#  —  y>     ik  —  k,     H  =  I,    ji  =  Id  =  il  =  0 

which  give,  by  §  46, 

0  =j*jk  =  lj  =  V  =  lj  =  lk  =  I*, 

jl  =  aui  ,      Id  =  a^i  , 


0  =r  /£//£  =  a3Jc  =  #34  =  kl  , 

and  there  is  no  pure  algebra  in  this  case.* 

[16].  The  defining  equations  of  this  case  are 

*y  =  y  »  &*'  =  *  >  y*  =  ^  =  ^  =  n  =  o  , 

which  give,  by  §  46, 

o=y»  =  x*=«=^ 

J7c  =  a23i,    jl=bHj,     7y'  =  dnl,     lk  —  cdc,     I*  =  JJ  , 


llej  =  dftdj,  =  c4343?,      pie  =  c&k  =  c43djc,      aZ3  =  l^d^  , 
0  =  a23  (c43  —  524)  =  Z>24  (524  —  ^24)  =  d32  (624  —  c^4)  =  d&  (c43  —  c?4)  =c43(c43  — 

There  are  two  cases: 

[161],  when  4,2  d°es  not  vanish; 
[162],  when  d3Z  vanishes. 

[161].  The  defining  equation  of  this  case  can  be  reduced  to 

<4  =  1  , 
which  gives 

43 


There  are  two  cases  : 

[1618],    when  d^  does  not  vanish  ; 
[1612],  when  d±  vanishes. 

[1612].  The  defining  equation  of  this  case  can  be  reduced  to 


which  gives 


*  In  relative  form,  i=A:A,j  =  A:B,  k  =  A:C,  l  =  D:A.     There  are  three  double  algebras  of 
theform(62).     [C.  S.  P.] 


36 


PEIRCE  :    Linear  Associative  Algebra. 


and  there  is  a  quadruple  algebra  which  may  be  called  (<74),  its  multiplication 
table  being 

i         j         If          I 


It 


I 


i 

j 

0 

0 

0 

0 

i 

j 

7c 

I 

0 

0 

0 

0 

Jc 

I 

This  is  a  form  of  quaternions* 

[1612].  The  defining  equation  of  this  case  is 

which  gives 


*  In  relative  form,  i  —  A:  A,  j—  A:B  ,  k  —  B:A,  l  =  B:B.  This  algebra  exhibits  the  general 
system  of  relationship  of  individual  relatives,  as  is  shown  in  my  paper  in  the  ninth  volume  of  the 
Memoirs  of  the  American  Academy  of  Arts  and  Sciences.  In  a  space  of  four  dimensions,  a  vector  may 
be  determined  by  means  of  its  rectangular  projections  on  two  planes  such  that  every  line  in  the  one  is 
perpendicular  to  every  line  in  the  other.  Call  these  planes  the  .A-plane  and  the  B-plane,  and  let  v  be 
any  vector.  Then,  iv  is  the  projection  of  v  upon  the  4-plane,  and  Iv  is  its  projection  upon  the  B-plane. 
Let  each  direction  in  the  A  -plane  be  considered  as  to  correspond  to  a  direction  in  the  .B-plane  in  such  a 
way  that  the  angle  between  two  directions  in  the  A  -plane  is  equal  to  the  angle  between  the  correspond 
ing  directions  in  the  J5-plane.  Then,  jv  is  that  vector  in  the  4-plane  which  corresponds  to  the  projection 
of  v  upon  the  B-plane,  and  lev  is  that  vector  in  the  5-plane  which  corresponds  to  the  projection  of  v  upon 
the  4-plane. 

Professor  Peirce  showed  that  we  may  take  i1  ,  j\  ,  fc:  ,  as  three  such  mutually  perpendicular  vectors 

in  ordinary  space,  that  i  —  ~  (1  —  ji,}  ,  ;  =  A  (^—  jfcj),  *=i{  —  j\  —  jfc,),  1  =  ^(1^.  jj).      [See,  also, 

Spottiswoode,  Proceedings  of  the  London  Mathematical  Society,  iv,  156.  Cayley",  in  his  Memoir  on  the 
Theory  of  Matrices  (1858),  had  shown  how  a  quaternion  may  be  represented  by  a  dual  matrix.]  Thus 
i,  ji  k,  I,  have  all  zero  tensors,  and  j  and  k  are  vectors.  In  the  general  expression  of  the  algebra, 
q  —  xi  -\-yj-\-zk-\-wl,  if  x  +  w  =  l  and  yz  —  x  —  a?2,  we  have  g2=g;  if  x  =  —  w  =  '/—yz,  then 
q2  =  0  .  The  expression  i  +  I  represents  scalar  unity,  since  it  is  the  universal  idemfactor.  We  have,  also, 


Tq  = 


—  yz  (i 


The  resemblance  of  the  multiplication  table  of  this  algebra  to  the  symbolical  table  of  §46  merits 
attention.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 


37 


and  there  is  a  quadruple  algebra  which  may  be  called  (7i4),  its  multiplication 
table  being* 

i          j         Jc          I 


If- 


i 

j 

0 

0 

0 

0 

0 

0 

If 

I 

0 

0 

0 

0 

0 

0 

[162].     The  defining  equation  of  this  case  is 


which  gives 


and  there  can  be  no  pure  algebra  for  it.f 

[2].  The  defining  equation  of  this  case  is 


in  =  0 . 


There  are  four  cases  : 


[21],  when  n  =  5; 
[22]  ,  when  n  =  4  ; 
[23],  when  n  =  3  ; 
[24],  when  w  =  2. 

[21].     The  defining  equation  of  this  case  is 


and  by  §60,  #=j,     i^  —  1^     i*=l. 

This  gives  a  quadruple  algebra  which  may  be  called  (i4),  its  multiplication  table 
being  J 

*  In  relative  form,  i  =  A:A,  j—  A  :B,  k  —  C:  A,  l  —  C:B.     [C.  S.  P.] 

tin  this  case,  i  —  A:A,  l  =  d4(B:B+C:C),j=A:Boi:  =  A:D,k=:C:Aor  =  E:A.    [C.  S.  P.] 
Jin  relative  form,  i=.A:B  +  B:C+C:D-}-D:E,  j—  A  :C+  B  :D  +  C  ':  E  ,   k=  A  :D-\-  B:E, 
l  —  A:E.     [C.  S.  P.] 


38 


PEIRCE  :    Linear  Associative  Algebra. 
(i4)      i          j         Jc  I 


J 


j 

Jc 

I 

0 

Jc 

I 

0 

0 

I 

0 

0 

0 

0 

o 

0 

0 

[28].  The  defining  equation  of  this  case  is 

;4  =  o, 

and  by  §59,  **=/,     i3  =  k. 

There  are  then,  by  §  64,  two  quadruple  algebras,  which  may  be  called  (j4)  and 
,  their  multiplication  tables  being  * 

0'4)     i         j         Je         I  and          (Jc4)     i         j         Jc          I 


I 


j 

Jc 

0 

0 

Jc 

0 

0 

0 

0 

0 

0 

0 

Jc 

0 

0 

Jc 

Jc 


j 

Jc 

0 

0 

Jc 

0 

0 

0 

0 

0 

0 

0 

Jc 

0 

0 

0 

[23].  The  defining  equation  of  this  case  is 

i3  =  0 , 
and  by  §  59 

*/  ' 

and  it  may  be  assumed  from  the  principle  of  §  63  that 
which  gives 


*  In    either    of    these    algebras,    i  —  A  :  B  +  B  :  C-f  C:D,    j=A:C+B:D,    k=  A  :  D;    and 
in  (j\)  l  =  A:E  +  E:D  +  A,:C,  while  in  (fcj  l  =  A:C.    [C.  S.  P.] 


PEIRCE:    Linear  Associative  Algebra.  39 

There  are  two  cases  :  [231],  when  il  =  k  • 

[232],  when  il=Q. 
[231].  The  defining  equation  of  this  case  is 

U  =  k  , 
which  gives 

y?=w=a?=o, 

Id  —  a3li  -f  63iy  4-  c3lk  4-  dsll  , 

Q  =  iki=:a31j+d3lk,     a31=0,     d3l  =  Q,     Id  =  bBlj  +  c31k  . 
So,  because  it*  =  0  ,       F  =  b3J  4-  c3&  , 

and  because  iU  =  0  ,     fc?  =  634y  +  c34&  ,     kj  =  Idi  =  cjd  =  bslc3lj  +  c|^  . 
0  =  kj'i  =  clJd  ,     c3l  =  Q  =  kj, 

Hi  =  Id  =  b31j  ,     li  =  b3li  +  buj  +  %&  ,      y  =  &  =  (&8i  + 
=  F  =  c37<^  =  c3  ,     t'tt  =  ^  =  b3j  ,      7A  -  J3i  +  543y  +  c 


But  M  contains  no  term  in  I  ,  so  that  d4  =  0  . 
M  =  il*  =  aj,     b3,=  a,,     c34  =  0, 

0  =  Z3  =  M  +  CAJ  ,        ^34  =  «4  =  0  =  U  ,        I*  =  bj  +  C4 

M  =tf  =  bnjl  =0,     0  =  M  =  681fe'  =  IU  =  b3l  =  Id  = 
U  =  541y  +  c41/6.  ;    fa  —  in  _  o 

There  are  two  cases  : 

[2  3  12],  when  c41  does  not  vanish  ; 
[2312],  when  c41  vanishes. 

[2  3  12].  The  defining  formula  of  this  case  is 

c«  ±  0  , 
and  if  ^  is  determined  by  the  equation 


we  have 


so  that  I  4-  ^?i  and  A;  +  pj  may  be  substituted  respectively  for  /and  &  ,  which  is 
the  same  as  to  make 


40 


PEIRCE  :    Linear  Associative  Algebra. 


and  there  are  two  cases : 

[2 3 13],    when  c4*  does  not  vanish ; 
[23122],  when  c4  vanishes. 

[22 13].  The  defining  equation  of  this  case  can  be  reduced  to 

P  =  7c. 
This    gives   a  quadruple    algebra  which  may  be  called  (74),   its  multiplication 

table  being  f 

(Z4)      i         j         Jc          I 


i 

j 

0 

0 

k 

j 

0 

0 

0 

0 

Jc 

0 

0 

0 

0 

I 

ck 

0 

0 

1 

[23122],  The  defining  equation  of  this  case  is 


*  7.  e.  the  new  c4  ,  or  what  has  been  written  ct  +pc41 .  In  all  cases,  when  new  letters  of  the  alpha 
bet  of  the  algebra  are  substituted,  the  coefficients  change  with  them.  [C.  S.  P.] 

fWhen  6  =  0,  c  rr  1  ,  we  have  Z  (i  —  I)  =  (i  —  I)  I  =  0  ;  so  that  by  the  substitution  of  *  —  I  for  i ,  the 
algebra  is  broken  up  into  two  of  the  form  (c2) .  Whenb=.0,  c  :£  1  ,  on  substituting  il=i—  I, 
j1=j—ck,k1  =  (c  —  l}2k,  *!  =  (<!  — 1)Z,  we  have  i\  =j\  ,  i1Z1  =  0,  Ili1~l21=kl;  so  that  the 
algebra  reduces  to  (r4) .  When  &  =.  1 ,  c  =.  0  ,  on  putting  i1  —  i  —  I ,  j\  —j  —  k  ,  we  have  if  =  ij  =  0  , 
Zij  —  jj  ,  I2  —  k  ;  so  that  the  algebra  reduces  to  (g4)  .  When  &r=  1 ,  c  4=  0  ,  on  putting  ^  =  s/c-1  (i  —  I)  , 
jj  —J+  (c  —  1)  fc,  we  have  if  =  Z2=A;,  ij  =  0,  Zij  — ^  ;  so  that  the  algebra  reduces  to  (pt). 
When  6(6  —  1)  (&c  +  &  —  1)  :£  0,  on  putting  ij  =  (1  — b)  6t—  (1  —  b)  Z,  J>i  =  (1  —  &)2(1—  &  —  &c)  ^, 
fci  =b2  (1  — b)  (1  —  b  —  be)  jf -b(l  —  b)(l  —  &  — c  +  c2b)A;,  ll  —b  (1  —  b)  i  —  bcZ ,  we  get  the  multipli 
cation  table  of  (o4).  When6(b  —  1)  ^i  0  ,  6c  +  b  =  l ;  on  putting  t1=b(t— Z),  j\=b2  (l  —  b}j—b2cJc , 
kl  =6(1  —  b  —  c)fc,  Zx  — bi  — Z,  we  get  the  following  multiplication  table,  which  may  replace  that  in 

the  text : 

(Z4)  i       j        Tc        I 


i 

3 
k 
I 

J 

0 

0 

/ 

0 

0 

0 

0 

0 

0 

0 

0 
0 

fc 

0 

0 

In  relative  form,  i  -A  :B+B:  C+  A:D,j=A:C,  k  =  A:E,  e=A:B  +  D:E.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 

There  are  two  cases: 

[231*21],  when  541  does  not  vanish; 

[231222],   when  641  vanishes. 
[231221].  The  denning  formula  of  this  case  is 


41 


There  are  two  cases  : 

[23P212],   when  c41  -f  1  does  not  vanish; 
[23P212],  when  c41  +  1  vanishes. 

[2312212].  The  defining  formula  of  this  case  is 

C41  +  1  t  0 

so  that 


c41  -f  1 


c41 


7  _ 

~ 


41 


so  that  the  substitution  of  4^   ,    "  ,       4lJ     ,  "C41 " ,  and  — -—  .  respectivelv,  for 

041  +  1  C41  -f-  1  C41  -f  1 

i,  y,  and  k,  is  the  same  as  to  assume 

c41  =  o,   z>41  =y, 

which  reduces  this  case  to  [2312]. 

[2312212],  The  denning  equation  of  this  is  easily  reduced  to 


This  gives  a  quadruple  algebra  which  may  be  called  (???4),   its   multiplication 
table  being 


« 

./ 

0 

0 

& 

.; 

0 

0 

0 

0 

A- 

0 

0 

0 

0 

z 

./-* 

0 

0 

0 

42 


PEIRCE  :   Linear  Associative  Algebra. 


The  substitution  of  i — I  and/ — It,  respectively,  for  i  and/  transforms  this 
algebra  into  one  of  which  the  multiplication  table  is  * 

(w4)    i         j         Jc          I 


0 

0 

0 

It 

0 

0 

0 

0 

0 

0 

0 

0 

j 

0 

0 

0 

[231222].  The  defining  equation  of  this  case  is 

This  gives  a  quadruple  algebra  which  may  be   called  (%),  its  multiplication 
table  being  f 

W    i         j        -k.      .1 


j 

0 

0 

& 

0 

0 

0 

0 

0 

0 

0 

0 

ck 

0 

0 

0 

[2312].  The  defining  equation  of  this  case  is 
which  gives 
so  that  the  substitution  of  I —  b^i  for  I  passes  this  case  virtually  into  [232], 

*i  =  A:B+C:D,  j  =  B:D,  k  =  A:C,  l  —  B:C.     [C.  S.  P.] 

t  In  relative  form,  i  =  A:B  +  B  :C+D  :E,  j=A  :C,  k=  A  :  E ,  l  —  B:E+cA:D.    When  c 
the  algebra  reduces  to  (#4).    [C.  S.  P.] 


PEIRCE  :   Linear  Associative  Algebra. 


43 


[232].  The  defining  equation  of  this  case  is 

il  —  0  , 
and  it  may  be  assumed  that 

Td  =  0  , 
Q=jl  =  kj  =  iW  —  tfi  =  ild  =  Hi  =  ilk  =  M  —  i7 

U  =  6  <?&         <Z 


There  are  two  cases  : 

[2321],  when  c41  does  not  vanish; 
[232*],  when  c41  vanishes. 
[2321],  The  defining  equation  of  this  case  is  easily  reduced  to 

U  =  /„•  , 
which  gives  0  =  lik  =  1?  =  HI  =  Id 


0  =  Z%  =  dfk  =  d\lk  =  d,  ,     Ik  =  a  J  =  Pi  , 

Z2  =  a4i  +  64y  +  c/-  , 

0  =  /3  =  ajc  -f  c4a4y  =  a4  =  Ik  . 

There  are  two  cases  : 

[23212],   when  c4  does  not  vanish  ; 
[23212],  when  c4  vanishes. 
[2321*].  The  defining  equation  of  this  case  can  be  reduced  to 


which   gives  a  quadruple  algebra  which  may  be  called  (o4),  its  multiplication 
table  being* 

(oj     i          j         *          Z 


j 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

i 

0 

0 

ty+ft 

*In  relative  form,  i  —  A  :E  +E  :  D  +  B  :  C,  j  —  A  :  D  ,  k=A:C,  l  —  A:B  +  B:C+bB:D. 
Wlien  &rz  0 ,  this  algebra  reduces  to  (r4).  When  6  =  —  1 ,  the  substitution  of  i  —  I  for  Z  reduces  it  to  [L). 
[C.  S.  P.] 


44  PEIRCE  :    Linear  Associative  Algebra. 

[23212].  The  defining  equation  of  this  case  is 

There  are  two  cases : 

[232121],  when  Z>4  does  not  vanish ; 

[232122],  when  ?>4  vanishes. 
[232121].  The  defining  equation  of  this  case  can  be  reduced  to 


This  gives  a  quadruple  algebra  which  may  be  called    (^4),  its   multiplication 


table  being* 


j 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

k 

0 

0 

* 

[232122].  The  defining  equation  of  this  case  is 

/2  =  0. 

This  gives  a  quadruple  algebra  which  may   be    called    (q±),   its   multiplication 
table  being  f 


j 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

& 

0 

0 

0 

*  In  relative  form,  t  =  .4  :B  + B  :D+ C:E,  j  =  A:D,  k  =  A:E,  l  =  A:C+C:D.     [C.  S.  P.] 
t  In  relative  form,  i=.A:C+C:D,j  —  A:D,  k  =  B:D,  I ~  B  :  C.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra.  45 


[2322].  The  defining  equation  of  this  case  is 

ft  =  b4ij 
and  we  have 

^  =  1>3j  +  c37f  +  d3l 
U  —  b      +  cc  +  dl 


P  =  bj  +  cjc  +  dj 

so  that  there  can  be  no  pure  algebra  in  this  case  if  541  vanishes,  *  and  it  may  be 
assumed  without  loss  of  generality  that 

U=j. 
There  are  two  cases  : 

[23221],  when  d3  does  not  vanish  ; 

[2323],    when  d3  vanishes. 
[23221].  The  defining  equation  of  this  case  can  be  reduced  to 


which  gives 


0  =  7j3  =  M=m=J*l= 


and  there  is  no  pure  algebra  in  this  case.f 

[2323].  The  defining  equation  of  this  case  is 

4  =  0, 
which  gives  0  =  1£  =  c3tf  =.  c3  ,     kz  =  b3j  , 

0  =  m  —  C34/^  H-  d*Jd  =  f?34  = 

0  =  1L*  —  G^  H-  djd  —  d^  = 

There  are  two  cases  : 

[23231],  when  b3  does  not  vanish; 

[2324],    when  b3  vanishes. 
[23231].  The  defining  equation  of  this  case  can  be  reduced  to 


which  gives 


this  case,  y,  k  and  Z,  might  form  any  one  of  the  algebras  (&3),  (c3),  (d3)  or  (e3).     [C.  S.  P.] 
t  The  case  is  impossible  because  fci  =  0  and  k2i  —j.     [C.  S.  P.] 


46 


PEIRCE  :   Linear  Associative  Algebra. 


so  that  /  —  b3ij  can  be  substituted  for  I  without  loss  of  generality,  which  is  the 
same  as  to  assume 


and  this  gives 


0  =  Z8  =  dp  =  dA  =  cjk  =  c,b43  =  Pk  =  c4  , 


so  that  there  is  no  pure  algebra  in  this  case.* 

[2324].  The  denning  equation  of  this  case  is 


which  gives 


#==0, 

ox=  #  =  **_=«%/=*, 

0  =  Id*  •=:  c3Jd  =.  cu  ,     Id  =.  bftj  , 


and  there  can  be  no  pure  algebra  if  c4  vanishes,  so  that  it  may  be  assumed, 
without  loss  of  generality,  that 

?  =  *, 

which  gives 

0  =  l3  =  Uc  =  Jd. 

This  gives  a  quadruple  algebra  which  may  be    called    (/-4),    its   multiplication 
table  being  f 

(n)   *       j       h       i 


j 

0 

0 

0 

0 

o 

0 

0 

0 

0 

0 

0 

i 

0 

0 

A- 

[24].  The  defining  equations  of  this  case  are 


Substituting  i  —  I  for  i ,  this  case  is,  i  =  B  :  D ,  j  =  A  :  D ,  k  =  A  :  C  +  C :  D ,  I  —  A  :  B      [C.  S.  P.  1 

':E.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra.  47 

and  it  may  be  assumed,  from  §§63  and  65,  that 

ij  =  It  =  — ji ,     il  =  U  =  0  , 
which  give 

0  =  ilt  =  Id  =  jit  =  //  =  Id  =  770 , 

so  that  there  is  no  pure  algebra  in  this  case.* 

QUINTUPLE  ALGEBRA. 
There  are  two  cases  : 

[1],  when  there  is  an  idempotent  basis  ; 
[2],  when  the  algebra  is  riilpotent. 

[1].  The  defining  equation  of  this  case  is 

There  are  eleven  cases  : 

[I2],   when/,  It,  I  and  m  are  all  in  the  first  group  ; 

[12],  when/,  It  and  I  are  in  the  first,  and  m  in  the  second  group  ; 

[13],  when/  and  It  are  in  the  first,  and  I  and  m  in  the  second  group ; 

[14],  when/  and  It  are  in  the  first,  I  in  the  second,  and  m  in  the  third  group  ; 

[15],  when/  is  in  the  first,  and  It,  I  and  m  in  the  second  group ; 

[16],  when/  is  in  the  first,  It  and  I  in  the  second,  and  m  in  the  third  group  ; 

[17],  when/  is  in  the  first,  It  in  the  second,  I  in  the  third,  and  m  in  the  fourth 

group  ; 

[18],  when  /,  It  and  I  are  in  the  second,  and  m  in  the  third  group  ; 
[19],  when/  and  /.;  are  in  the  second,  and  I  and  m  in  the  third  group  ; 
[101],  when/  and  k  are  in  the  second,  I  in  the  third,  and  m  in  the  fourth  group  ; 
[II1],  when/ is  in  the  second,  It  in  the  third,  and  I  and  m  in  the  fourth  group. 

[I2].  The  defining  equations  of  this  case  are 

*/  — /^  — y,     ik  —  Id  =  It ,     il  —  li  —  l,     im  —  mi  —  m. 

The  algebras  deduced  by  §69  from  algebras  (i4)  to  (r4)  may  be  named  (a5)  to  (/5), 
and  their  multiplication  tables  are  respectively 


i  =  —  A:C+B:E,j=A  :£+  C:E+  cD  \E,  k  =  A  :E,  I  —  —  A  :D  +  cB  :  E.    [C.  S.  P.] 


48 


i         j 


3 


PEiRCft  :    Linear  Associative  Algebra. 
k          I        m  (/>5)     *          j         k          I        m 


i 

j 

k 

I 

m 

j 

k 

I 

m 

0 

k 

I 

m 

0 

0 

I 

m 

0 

0 

0 

m 

0 

0 

0 

0 

(c5)      i          j          ~k          I         m 


i 

j 

k 

I 

m 

j 

k 

I 

0 

0 

~k 

I 

0 

0 

0 

I 

0 

0 

0 

0 

m 

/ 

0 

0 

0 

i 

j 

k 

I 

m 

j 

Jc 

0 

0 

I 

k 

0 

0 

0 

0 

I 

0 

0 

0 

0 

m 

Jc  —  1 

0 

0 

0 

m 


i 

j 

k 

I 

m 

j 

k 

I 

0 

0 

k 

I 

0 

0 

0 

I 

0 

0 

0 

0 

m 

Z 

0 

0 

I 

or       (<?5)     i          j 


k 
I 

m 


i         j         k         I        m 


i 

j 

~k 

/ 

m 

j 

Je 

0 

0 

I 

Tc 

0 

0 

0 

0 

I 

0 

0 

0 

0 

m 

aJc 
+U 

0 

0 

I 

I        m 


i 

j 

k 

I 

m 

j 

0 

0 

0 

I 

k 

0 

0 

0 

0 

I 

0 

0 

0 

0 

m 

k 

0 

0 

0 

1 

PEIRCE  :    Linear  Associative  Algebra. 


49 


m 


i 

j 

Jc 

1 

m 

j 

Jc 

0 

0 

1 

Jc 

0 

0 

0 

0 

1 

0 

0 

0 

0 

m 

al 

0 

0 

0 

Jc          I        m 


i 

i 

j 

Jc 

1 

m 

j 

j 

Jc 

0 

0 

0 

Jc 

Jc 

0 

0 

0 

0 

1 

1 

0 

0 

0 

0 

m 

m 

1 

0 

0 

Jc 

J 

Jc 

I 

m 


O's)     i         j         Jc          I        m 


i 

j 

i 
Jc 

1 

m 

j 

Jc 

0 

0 

0 

Jc 

0 

0 

0 

0 

1 

0 

0 

0 

0 

m 

Jc 

0 

0 

1 

Jc 


m 


i 

j 

Jc 

1 

m 

j 

Jc 

0 

0 

0 

* 

0 

0 

0 

0 

L 

6 

0 

0 

0 

n. 

1 

0 

0 

1+aJc 

Jc          I 


m 


i 

j 

Jc 

I 

m 

j 

Jc 

0 

0 

0 

K 

0 

0 

0 

0 

I 

0 

0 

0 

0 

m 

I 

0 

0 

0 

50  PEIRCE  :    Linear  Associative  Algebra. 

[12].  The  defining  equations  of  this  case  are 

ij  =  ji  =j,     ik  =  ki  =  k  ,     il  =  li  =  I  ,     tm  =  ?ri  ,     m^  —  0  , 

which  give,  by  §  46  , 

0  =  mj  =•  mJc  =  ml  =  m?, 

and  if  A  is  any  expression  belonging  to  the  first  group,  but  not  involving  i  ,  we 

have  the  form 

Am  —  am  , 

and  by  §  67,  A  is  nilpotent,  so  that  there  is  some  power  n  which  gives 

0  =  An  =  Anm  =  aAn~lm  =  anm  =  a  —  Am  , 
0  =jm  =  km  =1  Im  ; 


and  there  is  no  pure  algebra  in  this  case.* 

[13].  The  defining  equations  of  this  case  are 

ij  —ji  —  /,     Ht  —  ki  =  k,     il  =-1,     im  =•  m  ,     li  =  mi  =  0  , 
which  give,  by  §  46, 

0  =  Ij  =  Ik  =  P  =  Im  =  mj  =  mk  =  ml  =  m*  ; 
and  it  may  be  assumed  from  (c/3),  by  §  69,  that 

/  =  £,   y3  =  o. 

It  may  also  be  assumed  that 

jl  =  m  ,     whence  f     kl=  jm  =  0  . 

We  thus  obtain  a  quintuple  algebra  which  may  be  called  (7r5),  its  multiplication 
table  being  this  :  J 


*In  fact  i  and  m,  by  themselves,  form  the  algebra  (&2),  while  i,  j,  fc,  Z,  by  themselves  form 
one  of  the  algebras  (aj,  (&4),  (c4),  (d4),  the  products  of  m  withy,  k  and  I  vanishing.     [C.  S.  P.] 

fThisis  proved  as  follows:  0  =  J3l—j  2m  =  d2-.jl  +  e,  -,jm  —  d25e25l-{-  (d25  +  e225)m.     Thus  d2Se25 
=  0;  ord25=:0,  e25  =  0,  jm  —  U  —  Q.     [C.  S.  P.] 

:C,'k  —  A:C,l  —  B\D,  m  —  A:D.     [C.  S.  P.] 


PEIRCE  :   Linear  Associative  Algebra. 
i         j         Jc          I        m 


51 


i 

i 

Jc 

I 

m 

j 

Jc 

0 

m 

0 

Jc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[14].  The  defining  equations  of  this  case  are 

ij  =  ji  —  /  ,     ik  =  ki  =  k,     il  =•  I  ,     mi  =  m  ,     li  =  im  =•  0  , 

which  give,  by  §  46, 

0  =  jm  =  km  =  Jj=z  lit  =  P  =  ml  ==  m?. 

It  may  be  assumed  from  §  69  and  («3)  that 

/•=*,  y»  =  o, 

whence 

0  —  j7  =  /*;/  =  wy  =  w/^  =jlm  =  a45y  H-  6457^  =:  a45  =  &45  .     ??rz  =  c^Jc  , 

and  there  is  no  pure  algebra  in  this  case.* 

[15].  The  defining  equations  of  this  case  are 

ij  =  ji  =  j  ,     iJc  =  k  ,     il  =•  I  ,     im  =•  m  ,     ld  =  li-=.  mi  =•  0  , 
which  give,  by  §§  46  and  67, 

0  =y2  =  lg  =J<?=Jd  =  Jem  =  Ij  —  Ik  =  P  =  Im  =  mj  =  mk  =•  ml  =  m?. 
It  may  be  assumed  that  jlc  =  I,    jm  =  0  ,f 

whence,  jl=Q, 

and  there  is  no  pure  algebra  in  this  case.J 


*i—A:  A-\-B:B  +  C:C,  j—  A  :B  +  B:C,  k  —  A:C,  l  —  A:D.  m  =  cD:C.     [C.  S.  P.] 

t  We  cannot  suppose  j  k  =  fc  ,  because  j-k  =  Q.     We  may,  therefore,  put  I  for  jk.    Then.;7  =  0.    Then, 

Q-=.j2m'=^c25e25k-{-  (d25e25  +  c25)  Z  +  e2^m  .    It  follows  that  jwi  =  d2s^  an(i  substituting  m  —  d25fc  for 

m  ,  we  have  jm  =  0  .     The  algebra  thus  separates  into  (&2)  and  (e4).     [C.  S.  P.] 
:£,y--^l:5,  k  =  B:C,  l=A:C,  m  =  A:D.    [C.  S.  P.] 


52  PEIRCE  :    Linear  Associative  Algebra. 

[16].  The  defining  equations  of  this  case  are 


*)*  =y*  —  y  »     ^  —  &  ,     il  =  l,     mi  = 
which  give,  by  §§  46  and  67, 


=  li  =  im  =  0 


=     =m  =  g  =      = 
7cm  =  a35i 

and  it  may  be  assumed  that 


=  a^i  -f 


and  cZ23  cannot  vanish  in  the  case  of  a  pure  algebra,*  so  that  it  is  no  loss  of 
generality  to  assume 

...  .  /*='. 

which  gives 


There  are  two  cases  : 

[161],  when  «35  does  not  vanish  ; 
[162],  when  «35  vanishes. 
[161].  The  defining  equation  of  this  case  can  be  reduced  to 

%>  =  1  , 
which  gives  lm=.j,     7cm  =  i  +  &35y, 

and  i  +  635y  can  be  substituted  for  i,  and  this  gives  a  quintuple  algebra  which 
may  be  called  (/5),  of  which  the  multiplication  table  is 

ft)      i         j         Ik         I        m 


m 


< 

j 

7e 

I 

0 

i 

0 

I 

0 

0 

0 

0 

0 

0 

i 

0 

0         0 

0 

j 

m 

0         0 

0 

0 

b35l.  Hence  a, 5  =  0  and  either  c*2 8  or  63 5  =  0 , 
and  m  either  case  there  is  no  pure  algebra.  The  two  algebras  (ls )  and  (m5)  are  incorrect,  as  may  be  seen 
by  comparing  k  .  mk  with  km  .  k .  [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 
[162],  The  defining  equation  of  this  case  is 

«35  =  0  , 

b35j,     £m  — 0; 


53 


which  gives 


and  635  cannot  vanish  in  the  case  of  a  pure  algebra,  so  that  it  is  no   loss  of 
generality  to  assume 

km  =  j . 

This  gives  a  quintuple  algebra  which  may  be  called  (m5),  of  which  the  multipli 
cation  table  is 

*          j         & 


m 


i 

j 

* 

I 

0 

j 

0 

i 

0 

0 

0 

0 

0 

0 

j 

0 

0 

0 

0 

0 

m 

0 

0 

0 

0 

[17].  The  denning  equations  of  this  case  are 

*y  =y*  —  y  »  &  =  &  ,  u  =  ?  , 

which  give,  by  §§  46  and  67, 


=  17  =  ^  =  «M*  =  o  , 


0  = 

Jd  = 

0  = 

lid  = 

11cm  = 

7m?  = 


=y*  =jl=jm  =  kj  =tf  =  Ij  =  l*  =  lm  =  mf  — 


=  0  =  c^ml  =  c4344  ,     We  =  b^jk  =0  = 
=  c437?i2  =  c35c437?2  =  0  =  c43e5  , 
=  c^ld  ,      ((754  —  c35)  &34  =  0  ,     to8  =  e5to 
=  cl^ml  ,     (eB  —  rf54)  6?54  =  0  . 


=  c35to 


—  C35)%    =   0, 


54 


PEIRCE  :    Linear  Associative  Algebra. 


There  are  two  cases : 


[171],  whene5  =  l; 
[172], 


[171].  The  defining  equation  of  this  case  is 

771    ~~~  77? 

0  =  c43  =  Ik . 


which  gives 


There  can  be  no  pure  algebra  if  either  of  the  quantities  634  ,  c35  or  d^  vanish, 
and  there  is  no  loss  of  generality  in  assuming 


Jcl=j,     km  =  7c,     ml=l. 

This  gives  a  quintuple  algebra  which  may  be  called  (n$),  its  multiplication  table 
being 

(%;)     i         j         k          I         m 


J 


i 

j 

7c 

0 

0 

j 

0 

0 

0 

0 

0 

0 

0 

j 

7c 

I 

0 

0 

0 

0 

0 

0 

0 

I 

m 

[172].  The  defining  equation  of  this  case  is 

m2  — 0, 
0  =  c35  —  d$±  =  7»;7n  =  ml ; 


which  gives 


*  But  on  examination  of  the  assumptions  already  made,  it  will  be  seen  that  if  e5  is  not  zero,  and 
consequently  c43  =0  ,  the  algebra  breaks  up  into  two.  Accordingly,  the  algebra  (»i5)  is  impure,  for  i, 
j  ,  k  and  I ,  alone,  form  the  algebra  (/4),  "while  m  ,  I ,  k ,  j ,  alone,  form  the  algebra  (7t4),  and  im  =  mi 
—  0  .  [C.  S.  P.] 


PEIRCE  :    Linear    Associative  Algebra. 


55 


and  there  can  be  no  pure  algebra  if  either  534  or  c43  vanishes,  and  it  may  be 

assumed  that 

Id  =.j,     Uc  =  m . 

This  gives  a  quintuple  algebra  which  may  be  called  (05),  its  multiplication  table 

being  as  follows  :  * 

(o5)      i          j          Jc          I         m 


i 

i 

j 

7c 

0 

0 

j 

j 

0 

0 

0 

0 

Jc 

0 

0 

0 

j 

0 

I 

I 

0 

m 

0 

0 

m 

0 

0 

0 

0 

0 

[18].  The  defining  equations  of  this  case  are 

ij  =•  j ,     ik  •=•  k ,     il  =  I ,     mi  =  m  ,     ji  =.  Id  =  li  =  im  =  0 , 
which  give,  by  §  46  , 

But  if  A  is  any  expression  of  the  second  group, 

Am  =  ai ; 
which  gives 

0  =  Amj  =  aj  =  a  =  Am  =jm  =  7cm  =  Im , 

and  there  is  no  pure  algebra  in  this  case. 

[19].  The  defining  equations  of  this  case  are 

ij  =  j,     i7c  =  k ,     li  =  l,     mi  =  m  ,     il  ==.  im  =ji  =  Id  —  0  , 
which  give,  by  §  46, 

0  =y2  =jk  =  l?j  =  1<?  =  lj  =  Ik  =  1*  =  lm  =  mj  =  mk  =  ml  =  m9. 


—  B:B+D:D+F:F,  j  —  D:F,  k  =  B:C+D:E,  l  —  A:B  +  E:F,  m  =  A:C.    [C.  S.  P.] 


56  PEIRCE  :    Linear  Associative  Algebra. 

But  if  A  is  an  expression  of  the  second  group  and  B  one  of  the  third, 

AB  =  ai , 
which  gives 

0  =  ABj  =  aj  =  a  =  AB  =jl  =jm  =  Id  =.lm  , 

and  there  is  no  pure  algebra  in  this  case. 

[10'].  The  defining  equations  of  this  case  are 

ij  =  j,     ik  =.  k ,     li  =  I ,    ji  •=.  lei  =•  il  —  im  =  mi  •=.  0  , 
which  give,  by  §  46, 

and  it  is  obvious  that  we  may  assume 
We  have,  then, 


0  =  &&JI  =jml  =  c25  Id  = 

There  are  two  cases  : 

[lO'l],  when  a34  does  not  vanish  ; 
[10'2],  when  a34  vanishes. 

[lO'l].  The  defining  equation  of  this  case  can  be  reduced  to 

Id  =  i  , 
which  gives 


There  are  two  cases  : 

[10'P],   wheneB  =  l; 
[10'12],  when  e5  vanishes. 

[lO'l2].  The  defining  equation  of  this  case  is 

m*  •==.  m  ; 
and  we  assume 

jm  =j,     ml  =  I,     km  =  k, 

because  otherwise  this  case  would  coincide  with  a  subsequent  one.    We  get,  then, 

0  =JU  =  e&jm  =  e42  =  1j  ,         0  =jlk  =  e^jm  =  e43  =  Ik  , 
which  virtually  brings  this  case  under  [10'2].* 

*  This  does  not  seem  clear.     But  i  —  i2  =  klkl  =  0  ,  which  is  absurd.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra.  57 

[10'12J.  The  defining  equation  of  this  case  is 

m$   —    f\ 

m   — —  u  , 
which  gives 

0  =jmz  =  b^jm  =  625  =jm,     0  =  m*1  =  d^ml  =  a54  =  ml, 
0  =  km*  =  c^km  =  c35 ,     km  =  b^j ,     M  —  li  =  l  —  c^ml  =  0  , 

which  is  impossible,  and  this  case  disappears. 

[10'2].  The  defining  equation  of  this  case  is 

There  are  two  cases  : 

[10'21],  when  e5  =.  1  ; 
[10'22],   when  e5  vanishes. 

[10'21].  The  defining  equation  of  this  case  is 

mz  =:  m, 
and  if  we  would  not  virtually  proceed  to  a  subsequent  case,  we  must  assume 

jm  =j,     km  =  k,     ml  =  I, 
and  there  is  no  loss  of  generality  in  assuming 


so  that  there  is  no  pure  algebra  in  this  case.t 

[10'22].  The  denning  equation  of  this  case  is 

m*=0, 
which  gives 

0  =  mH  =  dMml.=  d^  =  ml  • 
and  we  may  assume 

%  =  0 , 
which  gives 

0  =jm?  =  bmjm  =  b,5  =jm,     0  =  km2  =  c35km  =  css ,     km 

0  =  e43m*  =  Ikm     =  bme,zm  =  535e42  ;  J 


*  In  this  case,  the  algebra  at  once  separates  into  an  algebra  between  j\  k ,  Z  and  m  .  and  three  double 

algebras  between  i  and  j ,  i  and  ft,  and  i  and  Z ,  respectively.     [C.  S.  P.] 

t  In  fact,  0  =  Iklk  =  e&m  =  e43=lk.    So  that  the  algebra  falls  into  six  parts  of  the  form  (&2 ) .    [C.  S.  P.] 
t  The  author  omits  to  notice  that  0  =  klk  =  e4  3km  =  e4 ,63  5 .     Thus,  either  km  =  0  or  lj  —  Ik  =  0 .    The 

algebra  (jof))  involves  an  inconsistency  in  regard  to  klk.     [C.  S.  P.] 


58 


PEIRCE  :    Linear  Associative  Algebra. 


and  we  have  without  loss  of  generality 

lj  =  0  ,     km  =  j,     Ik  =  m  . 
This  gives  a  quintuple  algebra  which  may  be  called  (ps),  of  which  the  multipli 

i          j          k          I         m 


cation  table  is 


k 


i 

j 

k 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

j 

I 

0 

m 

0 

0 

0 

0 

0 

0 

0 

mk  = 


[11'].  The  denning  equations  of  this  case  are 

ij  =y,     Id  =  k,    ji  =•  ik  =  il  =  im  =  li  =  mi  —  Q; 

which  give,  by  §  46, 

0  =j*  —  12  —  Id  =  km  =  Ij  =  mj  , 

jk  =  a23i  ,    jl  =  luj,    jm  =  &85(/,     kj  =  d^l  +  e&m  ,     Ik  =  c^k, 
There  are  two  cases  : 

[ll'l],  when  I  is  the  idempotent  base  of  the  fourth  group  ; 
[11'2],  when  the  fourth  group  is  nilpotent. 

[ll'l].  The  denning  equation  of  this  case  is 

l*  =  l. 
There  are  two  cases  : 

[ll'l2],   when  ?7i  is  in  the  second  subsidiary  group  of  the  fourth  group  ; 
[11'12],  when  m  is  in  the  fourth  subsidiary  group  of  the  fourth  group. 

[ll'l2].  The  defining  equations  of  this  case  are 

Im  —  m  ,      ml  =  0  ; 


PEIRCE  :    Linear  Associative  Algebra.  59 

which  give  0  —  m2  =jm*  =  b^jm  =  625  =jm  , 

0  =  mzk  =  c53mk  =.  c53  =•  mk  ; 

and  #23  cannot  vanish  in  a  pure  algebra,  so  that  we  may  assume 

jit  =  i , 
which  gives 

Jg'Ic  =  k  =  d3Zc±37t ,     jkj  =  j  =  d3zbzj ,      1  =  d32c43  =  dszbu , 
jl  =  jP  =  bujl ,     &224  =  &.J4  =  1 ,     Ik  =  Pk  =  c43lk ,     c|3  =  c43  =  1  =  c?33 , 
y/  —j ,     1k  =  1c,     Igl  =  Z  =  7^', 

and  there  is  no  pure  algebra  in  this  case.* 

[11/12].  The  defining  equations  of  this  case  are 

lm  •=.  ml  =  0 , 
which  give 

0  =  jlm  =  bujm  =  bubtoj  =  ^As.     0  =  Imk  —  c53//^  =  c43c537^  =  c43c53 , 
^7  =  <4g?  =  buJy  =  b^d^l  +  b^eazm  ,     %'  —  (732?  =  c4sJfj  =  c^d^l  +  c43c32m  , 
A^m  =  e32??i2  =  byjfj  =  b^d3Zl  +  bKe&m  ,     mkj  =  e3Zmz  =  c53Zy  =  c5342?  +  c^3e3Zm  , 
<^32 :=::::  bZ4d3%  —  c43ot32 ,      0  =  t>25ct32  =  653a32  =  O24e32  =  c43e32 .  y 

There  are  two  cases  : 

[11'121],  when  m  is  ideinpotent ; 

[11'122],   when  m  is  nilpotent. 
[11'121].  The  denning  equation  of  this  case  is 

m2  =  ?T?  , 
which  gives 

632  —  C53^32  :—  ^25e32  j 

and  it  may  be  assumed  that 

£24  =  0. 

But  if  the  algebra  is  then  regarded  as  having  I  for  its  idempotent  basis,  it  is 
evident  from  §  50  that  the  bonds  required  for  a  pure  algebra  are  wanting,  so 
that  there  is  no  pure  algebra  in  this  case.J 

*In  fact,  i,  j ,  fc,  I  form  the  algebra  (gr4),  and  Z,  m ,  the  algebra  (62).     [C.  S.  P.] 

t  The  last  equation  holds  by  I  68.     [C.  S.  P.] 

%  Namely,  d32z=0,  and  either  e32  — 1,  when  Z  forms  the  algebra  (Oj),  and  i ,  y ,  k ,  TO  the  algebra 
(gr4),  or  else  e32  =  0,  when  by  [13]  of  triple  algebra  a23  =0,  and  j  and  A;  each  forms  the  algebra  (62)  with 
each  of  the  letters  i,  I,  m.  [C.  S.  P.] 


60  PEIRCE  :    Linear  Associative  Algebra. 

[11'122].  The  defining  equation  of  this  case  is 

m2  =  0, 
which  gives 

0  =jm*  =  bKjm  =  b&j  =  625  =jm,     0  =  m*k  =  c^mk  =  c^k  =  c53  =  mk 


and  there  is  no  pure  algebra  in  this  case.  * 

[11/2],  The  defining  equation  of  this  case  is 

^=0, 

in  which  n  is  2  or  3.     We  must  then  have 

0  =  Im  =  ml  =  mz  , 
which  give 

0  =jls  =  b.2J(*  =  bjjl  =  bu  =jl  =jm  =  Ik  =  mk,      0  =  kjk  =  a,3k  =  «23  =  jk  , 
and  there  is  no  pure  algebra  in  this  case,  f 
[2].  The  defining  equation  of  this  case  is 

in  =  0  . 
There  are  five  cases  : 

[21],  when  n  =  6  ; 
[22],  when  n  =  5; 
[23],  when  n  =  4; 
[24],  when  n  =  3; 
[25],  when  7i  =  2. 

[21].  The  defining  equation  of  this  case  is 

i6  =  0  , 
and  by  §  60, 

P=j,     i3  =  k,     i*  =  l,     $  =  m. 

This  gives  a  quintuple  algebra  which  may  be  called  fe),  its  multiplication  table 
being 

Here,  m  forms  the  algebra  (6j)  ,  and  the  other  letters  form  (c/.J  .     [C.  S.  P.] 

t  Namely,  if  n  =  2  ,  j,  l  ,  fc  ,  form  the  algebra  (d3)  (second  form),  i,  y,  and  i  ,  ft  .  the  algebra  (6.),  and 
m  the  algebraic,).  Batif»=8,/,  k.  I  and  m  form  an  algebra  transformable  into  (,'4)  or  (fc4),  while  i, 
j  ,  and  z  ,  k  form,  each  pair,  the  algebra  (6a).  [C.  S.  P.] 


PEIRCB  :    Linear  Associative  Algebra. 
fe)     i          j          Jc          I          m 


61 


j 

Jc 

I 

m 

1 
0 

If 

I 

m 

0 

0 

I 

m 

0 

0 

0 

m 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[22].  The  defining  equation  of  this  case  is 

i5  =  0, 
and  by  §  59, 

There  are  then  by  §  64  two  quintuple  algebras  which  may  be  called  (r5)  and 
their  multiplication  tables  being 


j          Jc          I         m 


I        m 


Te 


j 

Jc          I 

0 

0 

Jc 

I         0 

0 

0 

I 

0         0 

0 

0 

0 

0         0 

0 

0 

I 

0         0 

0 

I 

Te 


j 

Jc 

I 

0 

0 

Jc 

I 

0 

0 

0 

{/ 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

0 

0 

0 

0 

[23].  The  defining  equation  of  this  case  is 

t4  =  0  ; 


and  by  §  59, 


t= 


62  PEIRCE  :   Linear  Associative  Algebra. 

and  it  may  be  assumed,  from  the  principle  of  §  63,  that 

8=0, 

which  gives 


0  =.jl  =  kl  =  iU  =  ilz  =  il 
li  =  c^k  +  dj.  +  6>41m  ,      I2  =  cjc  +  dAl  +  e^m  ,     Im  =  c^k  +  d45Z  +  e45™  • 

There  are  two  cases  : 

[231],  when  im  =  l-j 

[232],  when  im  =  0  . 
[231].  The  defining  equation  of  this  case  is 

im  =  I  , 
whence 

0  =  jm  =  km  =jmi  —jml  =  jm?  =  e41  =  e4  =  e45  , 
&  =  </47i  ,     0  =  K*  =  ^41?i3  =  dy*  =  dlU  =  d*  =  1j  =  Ik  , 

P  =  dif,     Q  =  l*  =  dJ3  =  dt,     K  =  cjc,     P  =  cjc,     Im^cJf.  +  dJ, 
imi  =  U  =  cdc  ,     mi  =  c4iy  +  c51^  +  cZ51?  ,     wy'  =  c41  (1  +  cZ51)  7f  ,      ?n^  =  0  , 
iwi?  =  P  =2  cjc  ,     ml  =  cj  H-  c54/^  +  e4t?  , 


0  =  m4  =  d45  ,     Kwi  =  ?2  =  c^&m  —  0  =  w?^  = 

0  =  mlm  —  d5Jm  =  duc^  ,      0  =  w?l  =  d^ml  =  cZ54  .* 

There  are  two  cases  : 

[2312],   when  c41  does  not  vanish  ; 

[2312],  when  c^  vanishes. 
[2  3  12].  The  defining  equation  of  this  case  is  reducible  to 

li  =  k. 
There  are  two  cases  : 

[2  3  13],    when  c45  does  not  vanish  ; 

[231*2],  when  c45  vanishes. 
[231s].  The  defining  equation  of  this  case  can  be  reduced  to 

Im  =  k, 
which  gives 


=  k-\-  d^k  +  d&k  =  k 


*  To  these  equations  are  to  be  added  the  following,  which  is  taken  for  granted  below  :  ml  =  mini 
C4gd61fc.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 

and  if  r  is  one  of  the  imaginary  cube  roots  of  —  1 ,  there  are  two  cases  : 

[2 3 14],     when  cZ51  =  r  ; 
[23132],  when<ZB1  =  — 1. 

[2 3 14].  The  defining  equation  of  this  case  is 

4,1  =  t , 


which  gives 


i  (m  —  c51Z)  =  /,     l(m  —  c51Z)  =  k , 
(m  —  cB1Z)  i  =  j  +  Vl ,     (m  —  cB1Z)/  =  (1  +  r)  &  , 
(w  —  c51?)  &  =  0  ,     (m  —  c51?)  I  =  vie , 
(m  -  cjf  =j  +  [CB  -  c51(l  +  r)]  k  +  (2r  -  1)  I ; 


so  that  the  substitution  of  m  —  c51?  for  wi  is  the  same  as  to  make 
There  are  two  cases : 


c51  =  0  . 


[2315],     when  c5  does  not  vanish  ; 
[23142],  when  c5  vanishes. 

[2315].  The  defining  equation  of  this  case  can  be  reduced  to 

<%==!•• 

There  is  then  a  quintuple  algebra  which  may  be  called  (£5),  its  multiplication 
table  being  * 

*  The  author  has  overlooked  the  circumstance  that  (t5)  and  (us)  are  forms  of  the  same  algebra.  If  in 
(tb)  weputij  =  i —  t2j ' ,  ji  —j —  2r2&,  kl  —  fc,  Zj  =.  —  v2k-\-  I ,  ml  =  —  r2J  +  m  ,  we  get  (1*5).  The  struc 
ture  of  this  algebra  may  be  shown  by  putting  i-L'=.  ri,  j\  =^l'2y,  fcj  =  —  A;,  Zx=r2J  —  rZ,  mj  i=  r«  —  m, 
when  we  have  this  multiplication  table  (where  the  subscripts  are  dropped): 

(«5)     i       j       k       I       m 


i 

i 

k 
I 

m 

j 

fc 

0 

k 

z 

k 

0 

0 

0 

k 

0 

0 

0 

0 

0 

rk 

0 

0 

0 

0 

rZ 

r2fc 

0 

0 

0 

In  relative  form,   i— A  :  B  +  A  :  C+  B  :  E+  C:  D  +  E:  (?,  j  =  A  :D+  A  :  E+  B  :G,  k  =  A:G, 
=  rA:E+C:G,  m^^A  :  B+  A  : F+  rC: E  +  D  :  G-  •  F:  G.    [C.  S.  P.] 


64 


PEIRCE  :    Linear  Associative  Algebra. 
i  j  k  I 


in 


k 


m 


j 

k 

0 

0 

i 

k 

0 

0 

0 

0 

0 

0 

0 

0 

0 

ft 

o 

0 

0 

«* 

j  +  r/ 

(i  +  r)& 

0 

r& 

j+k+ 

(2r—  1)1 

[23142].  The  defining  equation  of  this  case  is 

There  is  then  a  quintuple  algebra  which  may  be  called  (%„),  its  multiplication 
table  being 

(MB)       i  j  k  I  m 


k 


m 


j 

k 

0 

0 

I 

k 

0 

0 

0 

0 

0 

0 

0 

0 

0 

& 

0 

0 

0 

k 

y  +  rZ 

(i+r)ft 

0 

rk 

j  + 
(2t—  1)1 

[23132].  The  defining  equation  of  this  case  is 

fi    — i 

T   -    ,         .  ^51  •*•  j 

which  gives 

d5  =  Q,     i(m  —  c51?)  =  / ,     1(m  —  c51?)  =  k , 
(m  —  c5ll)i  =j  -  /,       (W  __  CBI?)/  =  _  ^  ,       (m  —  c5102  =y 
so  that  the  substitution  of  m  —  c51Z  *  for  m  is  the  same  as  to  make 

CM  =  0  . 


*  The  original  text  has  m  —  c51fc  throughout  these  equations,  but  it  is  plain  that  m  —  c,,l  is  meant. 

[C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 


65 


There  are  two  cases : 

[231321],  when  c5  does  not  vanish; 
[23P22],   when  c5  vanishes. 
[231*21].  The  defining  equation  of  this  case  can  be  reduced  to 


There  is  a  quintuple  algebra  which  may  be  called  (v5),  its  multiplication  table 
being  * 

K)     i          j          If          I         m 


Jc 


m 


j 

Jc 

0 

0 

I 

Jc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Jc 

0 

0 

0 

Jc 

j-l 

0 

0 

—  Jc 

j+Jc 

[231322].  The  defining  equation  of  this  case  is 

CB  =  0  . 

This  gives  a  quintuple  algebra  which  may  be  called  (w^),  its  multiplication  table 
being  * 

*  The  algebra  (VB)  reduces  to  (w5)  on  substituting  i,  =  i  +  ^j+  Jz .  j\  =j+  k,  A;1=fc.  l^-k  +  l. 

in>1  ~  8<?  ~*"  3-  z  +  TO  •     To  exhibit  the  structure  of  this  algebra,  we  may  put  p  and  p'  for  imaginary  cube 
roots  of  1,  and  substitute  in  (w*)  i-,  =i  +  //??i,   /,  —  (1  —  p}  ?4-  k~l-«/^RJ     j.  —or.     /   — /-i       ,/\  ,-_L  i. 

_  '•>   Ji  —  V-L        KjJ~T  Ot  ,    K! — £>«,<! — (I  —  P)J-\-K  — 

v  —  3Z ,  TOJ  =  i+  pm .     Then,  dropping  the  subscripts,  we  have  this  multiplication  table. 

i        j       k       I      m 


0       0 

0 

k 

) 

k   \    0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

k 

I 

k 

0 

0 

0 

In  relative  form,   i  -  f>'A  :B  +  P'C:F+  3PD  :  E,  j—  BpA  :  C  +  3p'D  :F,k  =  3A-D     1=  Sp'A  •  E 
+3pB:F.  m  =  pA:D  +  3p'B:C+pE:F.    [C.  S.  P.] 


66 


PEIRCE  :    Linear  Associative  Algebra, 
(w  )     i          j          k          I         m 


i 

j 
Jc 
I 
m 

j 

Jc 

0 

0 

I 

k 

0 

0 

0 

0 

0 

0 

0 

0 

0 

fc 

0 

0 

0 

fc 

)-l 

0 

0 

Jf. 

j+& 

[23P2].  The  defining  equation  of  this  case  is 

/ra  =  0, 
which  gives 

ml—  0,     mz  =  csk  +  dj,     m?i  —  djc  =  [1  +d51)&,     d,=  l+d^, 

and  c51  may  be  made  to  vanish  without  loss  of  generality. 
There  are  three  cases  : 

[231221],  when  neither  cZ51  nor  d51  +  1  vanishes  ; 
[23P22],   when  d51  +  1  vanishes  ; 
[231223],  when  d5l  vanishes. 

[23P21].  The  defining  formulae  of  this  case  are 

d51:£0,     ^i^-1- 
There  are  two  cases  : 

[2312212],   when  c5  does  not  vanish  ; 
[2312212],  when  c5  vanishes. 

[2312212].  The  defining  equation  of  this  case  can  always  be  reduced  to 


This  gives  a  quintuple  algebra  which  may  be  called  (x5),  its  multiplication  table 
being* 

*In  relative  form,  i  =  A  :B  +  A  :E+  B  :  D  +  D  :  F,  j—A:  D  +  B  :F.    k  —  A:F,    l  —  A:D. 
m=(l  +  a)  A  :B  +  A:C+A:E+B:D+C:D  +  D:F+E:F.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 
i  j  k  I  m 


k 


m 


j 

k 

0 

0 

I 

k 

0 

0 

0 

0 

0 

0 

0 

0 

0 

k 

0 

0 

0 

0 

j+al 

(1  +  a)k 

0 

0 

k+ 

(l+a)l 

[23P212].  The  defining  equation  of  this  case  is 

c5  =  0 . 

This  gives  a  quintuple  algebra  which  may  be  called  (//5),  its  multiplication  table 
being  * 

i          j  k-          I  m 


k 


m 


J 

k 

0 

0 

I 

k 

o. 

0 

0 

0 

0 

0 

0 

0 

0 

k 

0 

0 

0 

0 

j+v* 

(!+«)* 

0 

0 

(1  +  a)l 

[231222].  The  defining  equation  of  this  case  is 


which  gives 

There  are  two  cases: 


mi  =j  —  /,     mj  =  0 ,     m2  =  c5k . 

[2312221],  when  c5  does  not  vanish  ; 
[231223],    when  CB  vanishes. 


The  relative  form  is  the  same  as  that  of  (#5)  ;  omitting  from  m  the  terms  A  :  E  and  E :  F.    [C.  S.  P.] 


68 


PEIRCE  :    Linear  Associative  Algebra. 


[2312221].  The  defining  equation  of  this  case  can  be  reduced  to 

m*  =  k . 
This  gives  a  quintuple  algebra  which  may  be  called  (z5),  its  multiplication  table 

being  * 

(z5)      i          j          If          I         m 


i 

j 

k 

0 

0 

I 

j 

k 

0 

0 

0 

0 

k 

0 

0 

0 

0 

0 

I 

* 

0 

0 

0 

0 

m 

i-l 

0 

0 

0 

& 

[231223].  The  defining  equation  of  this  case  is 


This  gives  a   quintuple   algebra  which  may  be  called  (a«5),  its  multiplication 

table  being  f 

(fin5)     i          j          k          I         m 


i 

j 
Jc 
I 
m 

j          * 

0 

0 

I 

k          0 

0 

0 

0 

0          0 

0 

0 

0 

k         0 

0 

0 

0 

j-l       0 

0 

0 

0 

*In  relative   form,    i  —  A  :  B  +  B  :  C+  C:D  ,   j= .A  :  C+  B  :D  .    k=A:D.    l  =  A:C,    m  =  B:C 
+  A:E+E:D.     [C.  S.  P.] 

t  In  relative  form,  the  same  as  (z5),  except  that  m  —  B  :  C.     [C.  S.  P.] 


PEIRCE  :    Linear    Associative  Algebra.  69 

[231223].  The  defining  equation  of  this  case  is 

mi  =  j , 
which  gives 

0  =  (/  —  j)i  =  (m  —  i)i- 

so   that,   by  the  substitution  of  I — j  for  /  and  m  —  i  for  m,  this  case  would 
virtually  be  reduced  to  [232]. 

[2312].  The  defining  equation  of  this  case  is 

K  =  0, 

which  gives 

mj  =  0  ,      mim  =  ml  =  d^lm  ,     d45  =  0  ,     c54  =  d51c45 , 
m\  =  d5lml  =  c45k ,     c45  =  d5lcM  ,     ms  =  d5lm  =  d5ml ,     ^5(c54  —  c45)  =  0 . 

There  are  two  cases  : 

[23121],  when  d5  does  not  vanish  ; 
[23122],   when  d^  vanishes. 

[23121].  The  defining  equation  of  this  case  can  be  reduced  to 

d«  =  1 , 
which  gives 

C45  —  C54  5 

and  it  may  be  assumed  without  loss  of  generality  that 

c5  =  0.* 
There  are  two  cases  : 

[231212],   when  c45  does  not  vanish  ; 
[231212],  when  c45  vanishes. 

[231212].  The  defining  equation  of  this  case  can  be  reduced  to 

lm  =  ml  =  k , 
which  gives 

«t==l. 

There  are  two  cases  : 

[231213],     when  cB1  does  not  vanish  ; 
[2312122],  when  c51  vanishes. 

[231213].  The  defining  equation  of  this  case  can  be  reduced  to 


*  Namely,  by  putting  1 1  rz  c5  k  + 1  •  m  i  =  »i  —  c5i/ .     [C.  S.  P.] 


70 


PEIRCB  :    Linear  Associative  Algebra. 


This  gives  a  quintuple  algebra  which  may  be  called  (a&5),    its    multiplication 

table  being  * 

i          j          Tf          I         m 


k 


j 

k 

0 

0 

I 

Jc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

k 

Jc+l 

0 

0 

If 

i  +  i 

[2312122].  The  defining  equation  of  this  case  is 


This  gives  a  quintuple   algebra   which   may  be  called  (ac5),  its  multiplication 
table  being  f 

*The  structure  of  this  algebra  is  best  seen  on  making  the  following  substitutions:    Let  ^:  and  \)2 
represent  the  two  roots  of  the  equation  a?2  =  a?+l.     That  is,  fyl  rr  -  (l-\-*/  5)  and  fh—  ^  U  —  */  5)  • 


i).     Then,  we  have  the  multiplication  table  : 
i        j         k         I        m 


i 

3 
k 

I 
m 

3 

It 

0 

0 

ilhfc 

k 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

k 

iM 

0 

0. 

k 

I 

In  relative  form,  i  =  A:B  +  B  :C+  C:D+~  ^^  :G+  H:D,  j—A  :  C+B  :D.  k  =  A:D.  l=A:F 

D 


^2A  :H+G:D.     [C.  S.  P.] 
o 

t  On  making  the  same  substitutions  for  i  and  m  as  in  the  last  note,  this  algebra  falls  apart  into  two 
algebras  of  the  form  (63).     [C.  S.  P.] 


PEIBCE  :    Linear  Associative  Algebra. 
(ac5)     i         j         k         I         m 


71 


m 


] 

.;       ^ 

i 

0 

0 

I 

7c         0 

0 

0 

0 

0          0 

0 

0 

0 

0         0 

0 

0 

k 

I         0 

1 

0 

k 

/+! 

[231212].  The  defining  equation  of  this  case  is 

ml  =  Im  =  0 . 
There  are  two  cases  : 

[2312121],  when  c51  does  not  vanish  ; 
[2312122],   when  c51  vanishes. 

[2312121],  The  defining  equation  of  this  case  can  be  reduced  to 

%  =  1  • 

This  gives  a  quintuple  algebra  which  may  be  called  (ad^),    its   multiplication 
table  being* 

i          j         7c          I        m 


m 


j 

k 

0 

0 

I 

k 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

k  +  al 

0 

0 

0 

1 

*  In  relative  form,  t  —  A  :B+B:  C+C  :  D+E  :  F+aF:  G.  j—  A  :C+  B  :D  +  aE:G  ,  k  =  A:D. 
E:O,  m  =  A:C+E:F+F:G.     [C.  S.  P.] 


72  PEIRCE  :    Linear  Associative  Algebra. 

[231 2 122].  The  defining  equation  of  this  case  is 

This  gives  a  quintuple  algebra  which  may  be    called  (oe5),  its    multiplication 
table  being  * 

(ae§)     i         j          ~k          I        m 


j 

k 

0 

0 

I 

Tt 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

al 

0 

0 

0 

•J 

[23122].  The  denning  equation  of  this  case  is 

d,  =  Q. 
There  are  two  cases  : 

[231221],  when  c-45  does  not  vanish; 
[23123],    when  c45  vanishes. 

[23122!].  The  defining  equation  of  this  case  can  be  reduced  to 

1m  =  k, 
which  gives 

C45  —  °^51C45  )        °^51  =   1  • 

There  are  two  cases  : 

[2312212J,   whendB1  =  l; 
[2312212],  whend81=  — 1. 

[231 2212].  The  defining  equation  of  this  case  is 

...  .  <^  =  i. 

which  gives 


*  In  relative  form,  the  same  as  (od5 )  except  that  m  =  E:  F+  F:G.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra.  73 

There  are  two  cases : 

[23122!3],    when  c51  does  not  vanish  ; 
[23122122],  when  c51  vanishes. 

[23122!3].  The  defining  equation  of  this  case  can  be  reduced  to 

c51  =  1 . 

This    gives    a  quintuple  algebra  which   may  be  called  (c//5),  its  multiplication 
table  being* 

fa/s)     i          j          k          I        m 


m 


j 

Jc 

0 

0 

I 

Tc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

-k 

Jc+l 

0 

0 

~k 

j+cic 

*  To  show  the  construction  of  this  algebra,  we  may  substitute  il  =  i  -f  m ,  ji  —  2/+  (a  +  1)  k  +  21 , 
—  4& ,  ?!  =  2j  +  (a  -  1)  k  —  21 ,  ml—i  —  m.    This  gives  the  following  multiplication  table  : 


i 

3 
k 

I 
m 

j 

k 

0 

0 

2=1.. 

4    ; 

k 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

k 

-a+1fc 

~r' 

0 

0 

k 

I 

This  algebra  thus  strongly  resembles  (o65).      In  relative  foim,  t  —  A  :B  +  B  :  C+  C  :  D-f  A  :  G 


:D.     [C.  S.  P.] 


74  PEIRCE  :    Linear  Associative  Algebra. 

[23122122].  The  defining  equation  of  this  case  is 

cn  =  0. 
There  are  two  cases  : 

[231221221],  when  c5  does  not  vanish; 
[23122P22],    when  CB  vanishes. 

[231221221].  The  defining  equation  of  this  case  can  be  reduced  to 


This  gives  a  quintuple  algebra  which  may  be  called  (a#5),  its  multiplication  table 
being  * 

i          J         *          I 


m 


% 

j 

Jc 

I 

m 

j 

Jc 

0 

0 

I 

Jc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Jc 

I 

0 

0 

Jc 

j  +  Jc 

[231221222].  The  defining  equation  of  this  case  is 


This  gives  a  quintuple  algebra  which  may  be  called  (a/i5),  its    multiplication 
table  being  f 


*  On  substituting  t1=i+^t/  +  m,    m1=i+j—m,  this  algebra  falls  apart  into  two  of  the  form 

(63).     [C.  S.P.] 

t  On  substituting  il=i+m,  m1—i  —  m,jl—j+l,  l^^j  —  l,  this  algebra  falls  apart  into  two  of 
the  form  (63).     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 
(«7?5)     i         j         Jc          I         m 


75 


i 

j 
Jc 
I 
m 

j 

K 

0 

0 

I 

Jc 

0 

0 

o 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Jc 

I 

0 

0 

Jc 

j 

[2312212].  The  defining  equation  of  this  case  is 

which  gives 

There  are  two  cases : 

[23122121],  when  c51  does  not  vanish; 
[2312*12*],    when  c51  vanishes. 
[23122121].  The  defining  equation  of  this  case  can  be  reduced  to 

c51  =  1 . 

This  gives  a  quintuple  algebra  which  may  be  called  (<MS),    its   multiplication 
table  being  * 

(ai5)     i         j         Jc          I         'm 


Jc 


m 


j 

i 

0 

0 

I 

Jc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Jc 

fC  —  v 

0 

0 

-Jc 

j+cJc 

*In    relative    form,    t  =  A  :  C—  B  :  F+  C  :  E  +  D  :  G  +  E  :  G,   j  =  A  :  E+  C  :  G  .    k  =  A:O. 
—  A  :  F—  B:G.  m  =  A:B  +  A:D  +  B:E+C:  F+  aD  :  G  +  F :  G  .     [C.  S.  P.] 


76  PEIRCE  :   Linear  Associative  Algebra. 

[23122122].  The  defining  equation  of  this  case  is 

There  are  two  cases : 


mi  =•  —  ?. 


[231221221],  when  c5  does  not  vanish ; 
[23122123],    when  c5  vanishes. 

[2312*12*1].  The  defining  equation  of  this  case  can  be  reduced  to 

c5  =  1 .         - 

This  gives  a  quintuple  algebra  which  may  be  called   (o/5),    its   multiplication 
table  being* 

(0/5)     «          j         &          I        m 


j 

It 

0 

0 

I 

Jc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

ft 

—  I 

0 

0 

-& 

j  +  7c 

[23122123].  The  defining  equation  of  this  case  is 


This   gives    a  quintuple  algebra  which  may  be  called  (a&B),  its  multiplication 
table  being  f 


*  In  relative  form,  i  =  A  :  C+  C  :  E+  E  :  G  —  B  :F,  j—  A  :E+  C  :  G,  k 
=  A:B  +  B:E+C:F+F:G  +  A  :D  +  D:G.     [C.  S.  P.] 
t  In  relative  form,  t  =  A:C+  C  :  D-\-  D  :F—  B  \  E  ,  j  —  A  :D  -\-C\F.  k 
:D+C:E+E:F.     [C.  S.  P.] 


—  A:G,  l  —  A:F—B:G, 
=  A:F,  l=A:E—B:F, 


PEIRCE  :    Linear  Associative  Algebra. 
(a7c5)     i         j         7c          I        m 


77 


i 

j 

Jc 

0 

0 

I 

j 

7c 

0 

0 

0 

0 

Jc 

0 

0 

0 

0 

0 

I 

0 

0 

0 

0 

k 

m 

—  I 

0 

0 

~~~  K 

j 

[23123].  The  defining  equations  of  this  case  are 

ml  —  Ira  =  0 ,     mz  =  c5k . 
There  are  two  cases  : 

[231231],  when  cZ51  is  not  unity; 
[23124],    when  dm  is  unity. 

[23123!].  The  defining  equation  of  this  case  is 

.  .  *i*i. 

which  gives 

*  [(1  —  4i)  m  —  cB1y ]  =  (1  —  dm)  I  —  cB1& ,     i[(l  —  d5l)  I—  c517r ]  =  0  , 
[(1  —A)  l~  Cn&ji  =  0  ,      [(1  —  dn)  m  —  c5lj]i  =  d51  [(I—d5l)l  —  c5lk]  , 
[(1  —  4>i)  I  —  c5lk]  [(1  —  d,~]  m  —  c5iy]  =  0  , 
[(1  —  rfn)  m  —  c5J]  [(1  —  dm)  I—  c^]  =  0  , 
[(1  -  ^i)  m  -c51/]2  rz  (1  -  dWV ; 

so  that  the  substitution  of  (1  — d5l)  m  —  c51/  for  m,  and  of  (1  —  d5l)l  —  cB1&  for 
I,  is  the  same  as  to  make 

051  =   °  ' 

There  are  now  two  cases  : 

[23123!2],  when  c5  does  not  vanish  ; 
[2312312],  when  c5  vanishes. 

[23123!2].  The  defining  equation  of  this  case  can  be  reduced  to 

mz  =  k . 


78 


PEIRCE  :    Linear  Associative  Algebra. 


This  gives  a  quintuple  algebra  which  may  be   called  (a7B),    its    multiplication 
table  being* 

i         j        It         I        m 


m 


j 

Jc 

0 

0 

I 

Jc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

dl 

0 

0 

0 

Jc 

[2312312].  The  defining  equation  of  this  case  is 


This  gives  a  quintuple  algebra  which  may  be  called  (ow?5),  its  multiplication 
table  being 

(am5)   i         j         k         I         m 


m 


j 

Jc 

0 

0 

I 

7c 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

dl 

0 

0 

0 

0 

*In  relative  form,  i  —  A  :  B  +  B  :  C+C:D  +  dE:F,   j  —  A:C+B:D. 
m—A:E-\-B:F+E:D.    [C.  S.  P.] 


—  A\D,    l  —  A:F, 


PEIRCE  :   Linear  Associative  Algebra.  79 

[23124].  The  defining  equation  of  this  case  is 

There  are  two  cases : 

[231 241],  when  c51  does  not  vanish; 
[23125],    when  c51  vanishes. 

[23124!].  The  denning  equation  of  this  case  is  easily  reduced  to 

___  -i 
There  are  two  cases : 

[23124!2],   when  c5  does  not  vanish  ; 
[2312412],  when  c5  vanishes. 

[23124!2].   The  denning  equation  of  this  case  is  easily  reduced  to 

m2  =  k . 

This  gives  a  quintuple  algebra  which  may  be  called  (cm5),    its   multiplication 
table  being  * 

(an5)    i          j         Tc          I         m 


j 

Jc 

0 

o 

I 

7c 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

l  +  Tc 

0 

0 

0 

7c 

[2312412].  The  defining  equation  of  this  case  is 


This    gives   a    quintuple  algebra  which  may  be  called  (ao5),  its  multiplication 
table  being  f 


*In  relative  form,  i  =  A  :E+A  :B  +  B  :C+  C:D+E:F,   j—A  :C+B  :D+A:F, 

:F,m  =  A:C+A:E+E:D.     [C.  S.  P.] 

tin  relative  form,  i  —  A  :  B  +  B  :  C+  C  :  D  +  E  :  F,  j  —  A:C+B:D.    k  =  A:D 
A:C+A:E+B:F.     [C.  S.  P.] 


—  A'F 


80 


PEIRCE  :    Linear  Associative  Algebra. 
(ao5)    i  '      j          It  I         m 


Tf 

I 

m 


j 

Tf 

0 

0 

I 

If 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

l  +  Tf 

0 

0 

0 

0 

[23125].  The  defining  equation  of  this  case  is 

mi  =•  I. 
There  are  two  cases : 

[231252],  when  c5  does  not  vanish  ; 

[23126],    when  c5  vanishes. 

[23125!].  The  defining  equation  of  this  case  can  be  reduced  to 

m2  =  Jc. 

This  gives  a  quintuple  algebra  which  may  be    called  (op5),  its   multiplication 

table  being* 

i         j         1^          I         m 


J 
Jc 


j 

Tf 

0 

0 

I 

If 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

z 

0 

0 

0 

Tf 

*In  relative   form,    t  —  A  :  B  +  B  :  C'+  C  :  D  +  E  :  F,    j  =  A:C+B:D,    k  =  A:D,    l  —  A:F, 
:F+E:D.    [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 
[23126].  The  defining  equation  of  this  case  is 


81 


This  gives  a  quintuple  algebra  which  may  be  called  (aq^),   its   multiplication 
table  being 

(aq5)    i         j         7c          I         m 


J 


m 


j 

7c 

0 

0 

I 

7c 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

0 

0 

0 

0 

[232].  The  defining  equation  of  this  case  is 

im  =  0  ,* 

which  gives 

0  =  j m  =  km  , 
and  it  may  be  assumed  that 

li=  0. 
This  gives 

lj  =  Ik  =  0  =  iP  =  lzi  =  Urn  =  iml  =  mli  =  im . 

There  are  two  cases : 

[2321],  when  mi  =  l; 

[232s],   when  ww  =  0. 
[2321].  The  defining  equation  of  this  case  is 

mi  —  I, 
which  gives 

0  =  mj  =  mlc ,     Im  =  c^Jc  +  d^l  -\-  e^m , 

Imi  =  P  =  e^l ,     0  =  /4  =  e,J?  =  e^  =  72,     m2  =  cjc  +  dj  +  e,m , 
m*i  =  ml  =  ej,     0  —  m4/  =  e^mH  =  e5  =  ml;     0  =  ?w2  =  d^frw  =  ^45 . 


*  What  is  meant  is  that  every  quantity  not  involving  powers  of  i  is  nilfaciend  with  reference  to  i. 
Hence,  il  =  0  ,  also.     [C.  S.  P.] 


82 


PEIRCE  :   Linear  Associative  Algebra. 


There  are  two  cases : 

[23212],  when  c45  does  not  vanish; 

[23212],  when  c45  vanishes. 
[23212].  The  denning  equation  of  this  case  can  be  reduced  to 

lm  =•  /£,* 
which  gives 

m*  =  cjc,     (m  —  c5?)2  =  0  , 

so  that  the  substitution  of  m  —  csl  for  m  is  the  same  as  to  make 

This  gives  a  quintuple  algebra  which  may  be  called  (ar5),  of  which  the  multipli 
cation  table  is 

(«r5)     i         j         k          I          m 


j 

Tf 

0 

0 

0 

lc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Tc 

I 

0 

0 

0 

0 

[23212].  The  denning  equation  of  this  case  is 

lm  =  Q. 
There  are  two  cases : 

[232121],  when  d5  does  not  vanish; 

[232122],   when  d5  vanishes. 
[232121].  The  denning  equation  of  this  case  can  be  reduced  to 

d5  =  l. 
There  are  two  cases  : 

[2 32 12 12],   when  c5  does  not  vanish  ; 

[2321212],  when  c5  vanishes. 
[232121*].  The  defining  equation  of  this  case  can  be  reduced  to 

c5  =  1 . 


*  But  0  :=  im  r=  mim  •=.  lm  .    Thus,  this  case  disappears,  and  the  algebra  (or,,}  is  incorrect.     [C.  S.  P.] 


PEIECE  :   Linear  Associative  Algebra. 


83 


This  gives  a   quintuple  algebra  which  can  be   called  (c/,?3),   its  multiplication 
table  being* 

(as5)     i         j          Jc          I         m 


j 

k 

0 

0 

0 

Jc 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

0 

0 

0 

Jc  +  l 

[2321212].  The  defining  equation  of  this  case  is 

This  gives    a    quintuple    algebra  which   may  be  called  (W5),   its  multiplication 
table  being 

(at5)     i         j         Jf         I        m 


m 


j 

Jc 

0 

0 

0 

7c 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

0 

0 

0 

I 

[232122].  The  defining  equation  of  this  case  is 

mz  =  c5& . 
There  are  two  cases : 

[2321 2*1],  when  c5  does  not  vanish  ; 

[232123],    when  c5  vanishes. 


*In  relative  form,  i  —  A  :  B  +  B  :  C+  C  :  D  +  E  :  F,  j—A:C+B:D,    k  =  A:D, 
=  A:E+E:F+E:D.    Omitting  the  last  term  of  m  ,  we  have  (at.,).     [C.  S.  P.] 


=  A:F, 


84  PEIRCE  :  Linear  Associative  Algebra. 

[232122!].  The  defining  equation  of  this  case  can  be  reduced  to 


This  gives   a  quintuple  algebra  which  may  be  called 

table  being  * 

(au$)    i         j          If          I         m 


,  its  multiplication 


It, 


m 


j 

ft 

0 

0 

0 

If 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

0 

0 

0 

If 

[232123].  The  defining  equation  of  this  case  is 

m2  —  0 . 
This  gives   a    quintuple  algebra  which  may  be  called  (a«5),  its  multiplication 

table  being 

(av$)    i          j          If          I         m 


m 


j 

If 

0 

0 

0 

If 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

0 

0 

0 

0 

*In  relative  form,  i  —  A  :B+  B  :  C+  C:D,  j  —  A:  C+*B:D,  k  =  A:D,  l  —  E:D,  m  —  E\C 
+  A  :  F+  F :  D .    The  omission  of  the  last  two  terms  of  m  gives  (av5).     [C.  S.  P.] 


PEIRCB  :   Linear  Associative  Algebra.  85 

[2328].  The  defining  equation  of  this  case  is 

mi  =  0  , 
which  gives 

0  =  mj  —  mk  =  Imi  =•  m*i  , 

and  there  is  no  pure  algebra  in  this  case. 

[24].  The  defining  equation  of  this  case  is 

i3  =  0  , 
and  by  §  59  , 

**=/,  *y=y*=y*  =  o. 

There  are  three  cases: 

[241],  when  ik  —  I,  il  —  m  ; 

[242],  when  ik  =  l,  il  =  im  =  0  ; 
[243],  when  ik  =  il  =  im  =  0  . 

[241].  The  defining  equations  of  this  case  are 

ik  =•  I,     il  —  m  , 
which  give 

jk  —  m  ,     im  =jl  —jm  =  0  ,      0  =  iml  =  ml?  —  e54ml*  —  e54  ,    jk  =•  m  , 
tT*  =ml=  &My  ,     £2  =  6542i  -f  64y  +  e47n  ,      0  =  I3  =  b^m  +  e4ml  =  bu  =  ml  , 
im?  =  0  ,      m2  =  65y  +  e5m  ,     0  =  m3  =  e5m2  =  e5  , 
iwii  =  0  ,     mi  =  b^j  +  e^m  ,     mj  =  e^mi  ,     mi3  —  0  =  eB1  , 
i7i  =  mi  =  &B1y  ,     ?i  =  65ii  -f  ft^y  +  e41m  , 

fo7  =  lm=.  b51m  ,     0  =  ?3i7z  =  551  =  lm  =  mi  —  »wi7  =  -m2,     (li)  i  —  Ij  , 
i&2  =  lk  =  a3j  +  c3?  +  c^3m  ,     i?^  =  mk  =  c3m  ,     K/^  =  lz  =  a31m  , 
0  =r  77z^3  =  c|m  —  cs  =  mk  —  l$m  , 
ley  —  ^2  =  «31y  +  d31li  =  o31(l  -r  ^si)y  +  ^li^  , 

^i/  =  km  =  «31  (1  +  c?31)  m  ,      0  =  A?m  =  «31  (1  +  d3l)km  =  a31  (1  +  d31)  =  ^  , 
kj  =  <^wi  ,     0  =  &3  =  a3?  -j-  63w  +  dg?A;  =  as  =  b3  +  c^  =  b3kj  +  c^fc?  , 
Id  =  a31?  +  (631  +  cZ3cZ31)  w  ,      0  =  klk  =  a31lk  =  d3a3l  =  lid  =  a31lz  =  a3l  =  I2, 
0  =  ^3a31  =  b31d3 
*  0  =  In?  +  iki  +  M  = 
0  =  #i  +  /^  +  ^  = 
(k  +  pi)  i  =  b31j  H-  4^  +  %m  +  />/  =  (631  +  p  —pd3l)j  -\-  d31  (I  +pj)  +  e31m  , 


*  This  line  and  the  first  equation  of  the  next  can  be  derived  from  0=  (i+  jfc)3.     [C.  S.  P.] 


86  PEIRCE  :    Linear  Associative  Algebra. 

so  that  if  p  satisfies  the  equation 

the  substitution  of  k+pi  for  k  and  of  I  -\-  pj  for  /  is  the  same  as  to  make 

There  are  four  cases  : 

[241s],   when  neither  esl  nor  e3  vanishes  ; 
[2412],  when  e3l  does  not  vanish  but  e3  vanishes ; 
[2413],  when  e31  vanishes  and  not  e3 ; 
[2414],  when  esl  and  e3  both  vanish. 

[2412].  The  defining  equations  of  this  case  can  be  reduced,  without  loss  of 
generality,  to 

We  thus  obtain  a  quintuple  algebra  which  may  be  called  (aws),  its  multiplication 
table  being* 

(aw5)     i         j         k          I        m 


It 


j 

0 

I 

m 

0 

0 

0 

m 

0 

0 

H+w 

fm 

m 

0 

0 

Km 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[2412],  The  defining  equations  of  this  case  can  be  reduced  to 

_    1  _     /~k 

*  In  relative  form  i  =  A:B  +  B:D+vC:E+vE:F+G:F,  j  —  A:D-\-^C:F,  k  =  * i:C+B:E 
+  D  :  F+  A  :G+G:F,  1=  A  :  E+  B  :F,  m  —  A:F.  To  obtain  (aa?5),  omit  the  last  term  of  k.  To 
obtain  (cw/5),  omit,  instead,  the  last  term  of  i.  To  obtain  (oz5),  omit  both  these  last  terms.  [C.  S.  P.] 


PEIRCE  :   Linear  Associative  Algebra. 


87 


We  thus  obtain  a  quintuple  algebra  which  may  be  called  (ax5),  its  multiplication 
table  being 

(oa?B)    i         j         7c          I        m 


j 

0 

i 

m 

0 

0 

0 

m 

0 

0 

vl+m 

r2w 

0 

0 

0 

rm 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[2413],   The  defining  equations  of  this  case  can  be  reduced  to 

%  =  0 ,     e3  =  1 . 

We  thus  obtain  a  quintuple  algebra  which  may  be  called  (ay5),  its  multiplication 
table  being 

i         j         Jc          I         m 


k 


m 


j 

0 

I 

m 

0 

0 

0 

m 

0 

0 

tl 

r2m 

m 

0 

0 

Km 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[2414].  The  defining  equations  of  this  case  are 


We  thus  obtain  a  quintuple  algebra  which  may  be  called  («ZB),  its  multiplication 
table  being 


88 


PEIRCE  :    Linear  Associative  Algebra. 
(azs)     i         j         Jc          I        m 


j 

0 

I 

m 

0 

0 

0 

m 

0 

0 

rZ 

r2m 

0 

0 

0 

rm 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[242].  The  defining  equations  of  this  case  are 

ik  =.  I ,     il  =  im  =•  0 , 
which  give 

li  =  iki  =  OSLJ  +  c3ll ,     0  =  U3  =  csl  =  1$  =  Ij , 

U  — —  Ct3i  1  r&  —  IftiK  — —  I    —  1KI  ——  0^54  — —  C34  , 

iTem  —  Im  —  amj  +  c35? ,      0  =  Im3  =  c35 , 

0  =  imk  =  a53  =  c53 ,     0  =  mk3  =  e53 , 

There  are  two  cases : 

[2421],  when  %  does  not  vanish ; 
[2422],  when  e31  vanishes. 

[2421],  The  defining  equation  of  this  case  can  be  reduced  to 

Td  =.  m , 


e34  =  0 


PEIRCE  :    Linear  Associative  Algebra. 


89 


which,  by  the  aid  of  the  above  equations,  gives 

0  =  mi  =.  Idl  —  ml  —  1dm  =•  m?  ,     a3j  =  ik*  =•  kH  =  Ik  =  km  ,     0  =  Im  , 
b53j  =  kik  =  Id  =  mk  ,     0  =  iff  +  kik  +  l&i  =  2«3  +  b53  , 
0  =  k3  =.  a3  —  653  =  Id  —  km  =  mk  =  ml  ; 

and  if  ^  is  determined  by  the  equation 


k  -\-  pi  ,  I  -\-  pj  ,  and  ?n  +  pj   can   be  respectively  substituted  for  k,  I  and  m, 

which  is  the  same  thing  as  to  make 

63=0. 
There  are  three  cases  : 

[24212],   when  neither  d3  nor  e3  vanishes  ; 
[24212],  when  d3  vanishes  and  not  e3  ; 
[24213],  when  d3  and  e3  both  vanish. 

[24212].  The  denning  equation  of  this  case  can  be  reduced  to 


This    gives  a  quintuple  algebra  which  may  be  called  (&«5),  its  multiplication 

table  being* 

(ba5)     i          j         k          I        m 


k 


m 


j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

m 

0 

I  -\-ern 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

*In  relative  form,  i  =  A  :  B  +  B  :  C+  A  :  E,  j—  A  :  C,  fc=  D  :  B  +  E  :  F  -f  D  :  (?  +  eG  :  C+A  :E, 
1  =  A  :F,  m  —  D:C.  By  omitting  the  last  term  of  fc  and  putting  e  =  1  we  get  (6b5),  and  by  omitting  the 
last  two  terms  of  k  we  get  (&c5).  [C.  S.  P.] 


90  PEIRCE  :    Linear  Associative  Algebra. 

[24212].  The  defining  equation  of  this  case  can  be  reduced  to 


=  m. 


This  gives  a  quintuple  algebra  which  may  be  called  (665),    its   multiplication 

table  being 

(bb5)     i         j          Jc 


I        m 


m 


j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

m 

0 

m 

0 

0 

0 

o 

0 

0 

0 

0 

0 

0 

0 

0 

[24213].  The  defining  equation  of  this  case  is 


This   gives  a  quintuple    algebra  which  may  be  called  (6cB),   its  multiplication 
table  being 

(bc5)     i          j         7c          I        m 


i 

j 

0 

I 

0 

0 

j 

0 

0 

0 

0 

0 

le 

m 

0 

0 

0 

0 

I 

0 

0 

0 

0 

0 

m 

0 

0 

0 

0 

0 

PEIRCE  :    Linear  Associative  Algebra.  91 

[2422].  The  defining  equation  of  this  case  is 

%  =  0. 

There  are  two  cases  : 

[24221],  when  eB  does  not  vanish  ; 

[2423],    when  es  vanishes. 

jf 
[24221].  The  defining  equation  of  this  case  can  be  reduced  to 

1&  •=.  a3i  -f-  m  , 
which  gives 

Idk  =  kl  —  a3d3lj  ,     iJ$  =•  Ik  =  a3j  ,      J#i  =  a3j  -\-  mi  =•  d3lkl  —  a^d^j  , 
0  =  Wi  +  ik  H-  TdJc  —  a3  (d?3l  +  d3l  •+-  1)  ,     mi  =  a3  (d3ll  —  l)j  ,      ml  =  mik  =  0  , 
0  =  1$  •=.  a3l  +  mlt  —  a3ki  +  ^^  , 

mk  =  —  a%l  ,     km  =:  —  a3b3lj  —  a3d3ll  ,     Im  =  0  . 

There  are  two  cases  : 

[242212],    when  a3  does  not  vanish  ; 

[242212],  when  a3  vanishes. 
[242212].  The  defining  equation  of  this  case  can  be  reduced  to 

1$  =  i  -\-  m  , 
which  gives 

d3l  =.  v'  1  =  t  ,     lk=j,     mlc  =  —  I  , 

ki  =  —  km  =  b3lj  +  r?,     mi  =  (t2  —  1)  j"  ,     m2  =  —  Vj. 

There  are  two  cases  : 

[242213],    when  b3l  does  not  vanish  ; 

[2422122],  when  b3l  vanishes. 
[242213].  The  defining  equation  of  this  case  can  be  reduced  to 


This  gives  a  quintuple  algebra  which  may  be  called   (bd^),    its    multiplication 
table  being  * 

*  In  relative  form,  i  —  A  :  D  +  D  :  F+  B  :  E+  C  :  F,  j  -  A  :  F,  k  =  vA  :  B  +  vB  :  C+  D  :  E—  -D  :F 
-\-E\F,   i=A:E~-l-A:F-\-B:F.m  —  v2A:C—-A:D  —  B:E—C:F.     [C.  S.  P.] 


92 


PEIRCE  :    Linear  Associative  Algebra,. 
(bd5)       i  j  k  I  m 


k 


m 


j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

j+  rz 

0 

i  +  m 

tf 

-j-vl 

0 

0 

j 

0 

0 

(r2-l)y 

0 

j 

0 

-fj 

[242*1*2],  The  defining  equation  of  this  case  is 

Id  =  vl . 

This    gives    a    quintuple  algebra  which  may  be  called  (6eB),   its  multiplication 
table  being  * 

(bes)       i  j  k  I  m 


k 


j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

tl 

0 

i-\-m 

tf 

yl 

0 

0 

j 

0 

0 

(r2-l)j 

0 

i/ 

0 

-0 

[242*12].  The  defining  equation  of  this  case  is 

V  =  m , 
which  gives 

0  =  Id  —  Ik  —  km  =  mh  =  mz  —  tfi  —  mi  . 

There  are  two  cases  : 

[242*121],  when  b3l  does  not  vanish; 

[242*12*],  when  631  vanishes. 


*  On  adding  to  the  expression  for  k  in  the  last  note  the  term  —  A  :  C ,  we  have  this  algebra  in  relative 
form.     [C.  S.  P.] 


PEIRCE  :   Linear  Associative  Algebra. 
[242*121].  The  denning  equation  of  this  case  can  be  reduced  to 


93 


This  gives  a  quintuple  algebra  which  may  be    called    (6/5),   its    multiplication 
table  being  * 

i         j         k          I         m 


i 

j 
Jc 
I 
m 

j   j    o 

/         0 

i 

0 

0          0 

0         0 

0 

j+dl     0 

m         0 

0 

0         0 

0          0 

0 

0          0 

0          0 

0 

[2422122].  The  defining  equation  of  this  case  is 


This  gives  a  quintuple  algebra  which  may   be    called  (bg^),  its    multiplication 

table  being  f 

i          j         k  I         m 


i 

j 

0 

/ 

0 

0 

j 

0 

0 

0 

0 

0 

Jc 

dl 

0 

m 

0 

0 

I 

0 

0 

0 

0 

0 

m 

0 

0 

0 

0 

0 

*In   relative   form,   i  =  A  :  B+  B  :  C+  D  :  E,    j—A:C.     k  —  A: 
l  —  A:E,m  =  A:E-\-A:F.     [C.  S.  P.] 

tin  relative  form,   i  — A  :  B  +  B  :C+D  :  E,   j=A:C,   k  =  dA  \D  +  B  :  E+  B  :  F .  l  —  A:E. 
m=.A:F.     The  algebra  (ars)  is  what  this  becomes  when  d  —  0 .     [C.  S.  P.] 


94  PEIRCE  :    Linear  Associative  Algebra. 

[2423].   The  defining  equation  of  this  case  is 

e3  =  0, 
which  gives  * 

*  It  is  not  easy  to  see  how  the  author  proves  that  a3  =  0  .  But  it  can  be  proved  thus.  0  =  k3  = 
(asi  +  b.J+  d3l)k  =  a3l  +  a3d3j  . 

The  algebras  of  the  case  [2423]  are  those  quintuple  systems  in  which  every  product  containing  j  or  I 
as  a  factor  vanishes,  while  every  product  which  does  not  vanish  is  a  linear  function  of  j  and  I.  Any 
multiplication  table  conforming  to  these  conditions  is  self  -consistent,  but  it  is  a  matter  of  some  trouble 
to  exclude  every  case  of  a  mixed  algebra.  An  algebra  of  the  class  in  question  is  separable,  if  all 
products  are  similar.  But  this  case  requires  no  special  attention  ;  and  the  only  other  is  when  two 
dissimilar  expressions  U  and  V  can  be  found,  such  that  both  being  linear  functions  of  i  ,  k  and  m  , 
UV—VU—Q.  It  will  be  convenient  to  consider  separately,  first,  the  conditions  under  which 
UV—  FC7=0,  and,  secondly,  those  under  which  UV+VU—Q'.  To  bring  the  subjects  under  a 
familiar  form,  we  may  conceive  of  i  ,  k  ,  m  as  three  vectors  not  coplanar.  so  that,  writing 

U  =  xi  -}-  yk  -\-  zm  ,        V—  x'i-\-  y'k-{-  z'm  , 

we  have  x  ,  y  ,  z  ,  and  x'  ,  y'  ,  z'  ,  the  Cartesian  coordinates  of  two  points  in  space.  [We  might 
imagine  the  space  to  be  of  the  hyperbolic  kind,  and  take  the  coefficients  of  j  and  I  as  coordinates  of  a 
point  on  the  quadric  surface  at  infinity.  But  this  would  not  further  the  purpose  with  which  we  now 
introduce  geometric  conceptions.]  But  since  we  are  to  consider  only  such  properties  of  U  and  V  as 
belong  equally  to  all  their  numerical  multiples,  we  may  assume  that  they  always  lie  in  any  plane 

Ax  +  By  +  Cz  —  1  , 

not  passing  through  the  origin  ;  and  then  x  ,  y  ,  z  ,  and  x'  ,  y'  ,  z'  ,  will  be  the  homogeneous  coordinates 
of  the  two  points  U  and  V  in  that  plane.  Let  it  be  remembered  that,  although  i  .  k.  mare  vectors,  yet 
their  multiplication  does  not  at  all  follow  the  rule  of  quaternions,  but  that 


l.        mk  =  b35j-\-d35l  .        m2 
The  condition  that  UV—  VU—Q  is  expressed  by  the  equations 

(b11  —  bsl)(xy'  —  x'y)  +  (bl5  —  b5l)(xz'  —  x'z}  +  (b35—  &53)  (yz'—y'z)  —  Q, 
(d13—d3l)(xy'  —  x'y)  +  (dl5  —  d^}(xz'  —  x'z)-\-  (d35  —  d53)(yz'  —  y'z)  =  Q. 

The  first  equation  evidently  signifies  that  for  every  value  of  U.  Fmust  be  on  a  straight  line,  that  this 
line  passes  through  U.  and  that  it  also  passes  through  the  point 


The  second  equation  expresses  that  the  line  between  U  and  V  contains  the  point 

Q=(d35  -d53)i+  (d51  —  dl5)k  +  (dl3—d31}m. 

The  two  equations  together  signify,  therefore,  that  C7and  Frnay  be  any  two  points  on  the  line  between 
the  fixed  points  P  and  Q.    Linear  transformations  of  j  and  I  may  shift  P  and  Q  to  any  other  situations 
on  the  line  joining  them,  but  cannot  turn  the  line  nor  bring  the  two  points  into  coincidence. 
The  condition  that  UV+  VU=  0  is  expressed  by  the  equations 

2b1a-a;'+  2b3yy'+  2b5zz'  +  (&is  +  b31)(xy'+  x'y]  +  (615  +  b5l}(xz'+  x'z]  +  (b35  +  bsa)(y*+y'z)  =  0, 

2d^xx'+  2d3yy'-{-2d-}zz'  -f  (dls  -f  dsl)(a#'+  afy)  +  (di  5  +  d-ol}(xz'+  x'z)  +  (d35  +  d53)(yz'+y'z)  =  0  . 

The  first  of  these  evidently  signifies  that  for  any  position  of  Fthe  locus  of  U  is  aline  ;  that  U"  being  fixed 

at  any  point  on  that  line,  V  may  be  carried  to  any  position  on  a  line  passing  through  its  original  position  : 

and  that  further,  if  U  is  at  one  of  the  two  points  where  its  line  cuts  the  conic 

b522  +  (Z>18  +  b81)  xy+  (615  +651)  xz+  (&85 


PEIRCE  :    Linear  Associative  Algebra. 


95 


0  =  l£i  =  a3j  =  a3  =  Z/b  =  wiZ  =  M 
0  =  Tcmk  =  a?  = 


then  F  may  be  at  an  infinitely  neighboring  point  on  the  same  conic,  so  that  tangents  to  the  conic  from 
V  cut  the  locus  of  C7"  at  their  points  of  tangency  .  The  second  equation  shows  that  the  points  U  and  V 
have  the  same  relation  to  the  conic 


3y2  +  d-,z2  +  (d13  +  d31)  xy+  (dls  +  d-al)  xz+  (d35  +d5B)  yz~  0  . 

These  conies  are  the  loci  of  points  whose  squares  contain  respectively  no  term  in  j  and  no  term  in  I  . 
Their  four  intersections  represent  expressions  whose  squares  vanish.  Hence,  linear  transformations  of 
j  and  I  will  change  these  conies  to  any  others  of  the  sheaf  passing  through  these  four  fixed  points.  The 
two  equations  together,  then,  signify  that  through  the  four  fixed  points,  two  conies  can  be  drawn 
tangent  at  U  and  V  to  the  line  joining  these  last  points. 

Uniting  the  conditions  of  UV—  VU—  0  and  UV-\-  VU—  0  .  they  signify  that  U  and  V  are  on  the 
line  joining  P  and  Q  at  those  points  at  which  this  line  is  tangent  to  conies  through  the  four  fixed  points 
whose  squares  vanish.  But  if  the  algebra  is  pure,  it  is  impossible  to  find  two  such  points  ;  so  that  the 
line  between  P  and  Q  must  pass  through  one  of  the  four  fixed  points.  In  other  words,  the  necessary 
condition  of  the  algebra  being  pure  is  that  one  and  only  one  nilpotent  expression  in  i  .  k.  m  .  should  be 
a  linear  function  of  P  and  Q  . 

The  two  points  P  and  Q  together  with  the  two  conies  completely  determine  all  the  constants  of  the 
multiplication  table.  Let  S  and  T  be  the  points  at  which  the  two  conies  separately  intersect  the  line 
between  P  and  Q  .  A.  linear  transformation  ofj  will  move  P  to  the  point  pP-\-  (1  —  p)  Q  and  will  move 
S  to  the  point  pS-{-  (1  —  p)  T  ,  and  a  linear  transformation  of  I  will  move  Q  and  T  in  a  similar  way.  The 
points  P  and  S  may  thus  be  brought  into  coincidence,  and  the  point  Q  may  be  brought  to  the  common 
point  of  intersection  of  the  two  conies  with  the  line  from  P  to.  Q  .  The  geometrical  figure  determining 
the  algebra  is  thus  reduced  to  a  first  and  a  second  conic  and  a  straight  line  having  one  common  intersec 
tion.  This  figure  will  have  special  varieties  due  to  the  coincidence  of  different  intersections,  etc. 

There  are  six  cases  :  [1],  there  is  a  line  of  quantities  whose  squares  vanish  and  one  quantity  out  of 
the  line  ;  [2],  there  are  four  dissimilar  quantities  whose  squares  vanish  ;  [3],  two  of  these  four  quantities 
coincide  ;  [4],  two  pairs  of  the  four  quantities  coincide  ;  [5].  three  of  the  four  quantities  coincide  :  [6],  all 
the  quantities  coincide. 

We  may,  in  every  case,  suppose  the  equation  of  the  plane  tobex-\-y-{-z=.l. 

[1].  In  this  case,  the  line  common  to  the  two  conies  may  be  taken  as  y  ==  0  ,  and  the  separate  lines  of 
the  conies  as  z  :=  0  and  x  —  0  ,  respectively.  We  may  also  assume  2P  -=.x-\-y  and  2Q  —  x  +  z  .  We 
thus  obtain  the  following  multiplication  table,  where  the  rows  and  columns  having  j  and  I  as  their 
arguments  are  omitted  : 

i  k         m 


0 

3Z        -j 

—  I 

0       Bj+l 

J 

i-j     o 

[2].     In  this  case,  we  may  take  k  as  the  common  intersection  of  the  two  conies  and  the  line,  i ,  m , 
and  i  —  k  -f-  in  as  the  other  intersections  of  the  conies.     We  have  Q  =  k  .  and  we  may  write 

—  rp  —  rq}k  +  rgm  . 


T—rP-}-  (\—r}Q  — 
We  thus  obtain  the  following  multiplication  table  : 


PEIRCE  :    Linear  Associative  Algebra. 


There  are  two  cases  : 


[24231],  when  d5  does  not  vanish  ; 
[2424],    when  d$  vanishes. 

[24231].  The  defining  equation  of  this  case  can  be  reduced  to 


which  gives 
and  if 


i  (k  +  bsi  +pm)  =  I  +  bj  =  mz  ; 


3) 


m 


0 

q(q  +  l)j+rq(rq  —  l)l 

[-2-p(p-3)  +  q(q+l)]j 
+  [—  2  —  rp(rp—  l}  +  rq(rq  —  \}}l 

q(q  —  $)j+rq(rq  —  l)l 

0 

—  p(p  —  B)j—rp(rp  —1)1 

[2-p(p+l)  +  a(g-8)]y  + 
[2  —  rp  (  rp  —  1)  +  rq  (rq  —  1)J| 

—  p  (p+l)j  —  rp  (rp—  1)  I 

0 

[3].  Let  k  be  the  double  point  common  to  the  two  conies,  and  let  i  and  m  be  their  other  intersections. 
Then  all  expressions  of  the  form  ku  +  t/fc'are  similar.  The  line  between  P  and  Q  cannot  pass  through 
k ,  because  in  that  case  all  products  would  be  similar.  We  may  therefore  assume  that  it  passes  through 
i .  Then,  we  have  Q  =  i .  we  may  assume  S  z=  P  =  i  — -  Jc+m  ,  and  we  may  write  T—  rP  +  (1  —  r)  Q 
—  i  _  rfc  -\-  rm  .  The  equation  of  the  common  tangent  to  the  conies  at  k  may  be  written  hx+  (I  —  h)z  —  Q. 
Then  the  equations  of  the  two  conies  are 

hxy  -f  xz  +  (1  —  h)yz  =  0 , 
hxy  +  (h  +  f  —  hr)xz  +  (1  —  h)yz  =  0 . 

We  thus  obtain  the  following  multiplication  table  : 

i  k  m 


m 


[4].    In  this  case  we  may  take  i  and  m  as  the  two  points  of  contact  of  the  conies,  k  as  P,  and 
t  —  k  +  m  as  71.     Then  writing  the  equations  of  the  two  tangents 

gy  +  z  —  0  .         x  +  hy  —  0  . 

g^  +  a?z  +  %2  —  0  . 
h  —  1)  ?/2  -f  gxy  +  xz  +  hyz  —  0  . 

and  the  multiplication  table  is  as  follows  : 


the  two  conies  become 


PEIRCE  :   Linear  Associative  Algebra. 


97 


the  substitution  of  1&  -f  ^  +  pin  for  /;  and  Z  +  Z>5/  for  ?  is  the  same  as  to  make 

&5  =  rfg  =  0  . 

o  o 

This  gives 

There  are  two  cases  : 

[242312],   when  b3  does  not  vanish  ; 
[242*12],  when  13  vanishes. 

[2428!1].  The  denning  equation  of  this  case  can  be  .reduced  to 

«*=,-. 


(g  +  h  —  i)  I 

ffl'+to+D* 

2? 

gj  -\-  (g  —  1)  I 

0 

V+(»+D. 

V 

V+(»-«. 

0 

[5].    In  this  case,  we  may  take  k  as  the  point  of  osculation  of  the  conies  and  i  as  their  point  of  inter 
section.     The  line  between  P  and  Q  must  either,  [51],  pass  through  fc,  or,  [52],  pass  through  i  . 
[51].     We  may,  without  loss  of  generality,  take 


and  the  equations  of  the  two  conies  are 

22  +  rxz  =  0  ,        rxy  +  2qxz  +  2yz  =  0  . 
Then,  the  multiplication  table  is  as  follows  : 

i  k          m 


0 

0 

ql 

rl 

0 

I 

rj+ql 

I 

j 

[52].    We  have   Q  —  i,   we  may  take  T7—  m,  and  we  may  assume 
Then,  we  may  write  the  equations  of  the  two  conies, 


—  2i  —  m  and  &13  -f-  &si  —  1 


2zz  +  xy  +  xz 
—  ra-?/  -f-  (2  —  r)  xz  -}-  ?'2?/2  =  0  . 


We  thus  obtain  the  following  multiplication  table  : 


98  PEIRCE  :    Linear  Associative  Algebra. 

This    gives  a  quintuple  algebra    which    can    be  called  (&7/5),  its  multiplication 
(Mis)      i         j          ^          I          m 


table  being  * 


j 

o       / 

0 

0 

0 

0         0 

r 

0 

0 

n» 

o       j 

0 

ej+*l 

0 

0         0 

I 

0 

0 

A 

o  L5 

r/7      ° 

I 

0 

—  rt 

j_(r-2)Z 

2/-H 

0 

(r-2)j 
+  (r2  +  l)Z 

j-(r-2)Z 

(r-2)j 
+  (r*-l)Z 

V 

m    j  —  (r  - 

[6].  The  conies  have  but  one  point  in  common.     This  may  be  taken  at  k .     We  have  Q  =  k ,  we  may 
take  T—i  and  2P  —  2S  —  i+k.     We  may  also  take  bl—  —  l.     Then  the  equations  of  the  two  conies 

may  be  written 

1   2xii  -\-  4:Qxz  -4-  2n/z  —  0 

r)  xz  +  2ryz  =.  0 . 


We  thus  find  this  multiplication  table  : 


—  j 

j+l 

(2q  —  l)j+2(q  +  r  —  p)l 

j+l 

0 

(r+l)j+fi 

(2q+l)+2(q+r+p)l 

(r-l)/+ri 

l>J4-(4-fpr*)l 

If  this  analysis  is  correct,  only  three  indeterminate  coefficients  are  required  for  the  multiplication 
tables  of  this  class  of  algebras.  [C.  S.  P.] 

*  See  last  note.  I  do  not  give  relative  forms  for  this  class  of  algebras,  owing  to  the  extreme  case 
with  which  they  may  be  found.  [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 


99 


[242312].  The  defining  equation  of  this  case  is 

I*  =  0 . 
There  are  two  cases : 

[2423121],  when  &31  does  not  vanish  ; 
[242812»],    when  631  vanishes. 

[242*121].  The  defining  equation  of  this  case  can  be  reduced  to 

• 

bsl  =  1 . 

This  gives  a  quintuple  algebra  which  may   be    called  (&/5),  its   multiplication 

table  being 

(M5)      i         j          Jc          I         m 


i 

j 

0 

« 

0 

0 

j 

0 

0 

o 

0 

0 

~k 

j  +  al 

0 

0 

0 

bj+cl 

I 

0 

0 

0 

0 

0 

m 

a'j 
+  VI 

° 

<0 

+d'l 

0 

I 

[2423128].  The  defining  equation  of  this  case  is 


There  are  two  cases  : 


[24231221],  when  &51  does  not  vanish  ; 
[2423123],     when  651  vanishes. 


[24231221].  The  defining  equation  of  this  case  can  be  reduced  to 


This    gives   a    quintuple    algebra  which  may  be  called  (lj\),  its  multiplication 


table  being 


100 


PEIRCE  :    Linear  Associative  Algebra. 
i         j          k          I         m 


j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

(ll 

0 

0 

0 

fy'+cZ 

0 

0 

0 

0 

0 

j+M 

0 

Vj+il 

0 

I 

[2422123].  The  defining  equation  of  this  case  is 


which  can  always,  in  the  case  of  a  pure  algebra,  be  reduced  to 

mi  •=.  I  . 

This  gives   a    quintuple  algebra  which  may  be  called  (&&5),  its  multiplication 
table  being 

(bks)     i         j         It         I         m 


j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

al 

0 

0 

0 

#  +  cZ 

0 

0 

0 

0 

0 

I 

0* 

a'j 

+  VI 

0 

I 

[2424].  The  defining  equation  of  this  case  is 


PEIRCE  :    Linear  Associative  Algebra. 
and  it  can  be  reduced  to  [24231]  unless 

whence  it  may  be  assumed  that 

and  since 

when 


101 


k  +  lif  =  0 , 
tf  =  0 . 


it  may  also  be  assumed  that 

There  are  two  cases  : 

[24241],  when  b3l  does  not  vanish; 
[2425],    when  b3l  vanishes. 

[24241].  The  defining  equation  of  this  case  can  be  reduced  to 


This   gives   a  quintuple    algebra  which  may  be  called  (5?5),  its  multiplication 

table  being 

i         j        If         I       m 


m 


j 

0 

z 

0 

0 

0 

0 

0 

0 

0 

j-l 

0 

0 

0 

aj-\-bl 

0 

0 

0 

0 

0 

3 

0 

aj+U 

0 

y 

[2425].  The  defining  equation  of  this  case  is 

ki  =  —  L 


There  are  two  cases : 


[24251],  when  635  does  not  vanish ; 
[2426],    when  J35  vanishes. 


102 


PEIRCE  :    Linear  Associative  Algebra. 


[24251].  The  defining  equation  of  this  case  can  be  reduced  to 

&S5  =  1  • 

This  gives  a   quintuple  algebra  which  may  be  called  (6w?5),  its  multiplication 
table  being  * 

If          I         in 


m 


j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

V 

0 

0 

0 

j+al 

0 

0 

0 

0 

0 

j 

0 

bj  —  cd 

0 

<9 

[2426].  The  defining  equation  of  this  case  is 


There  are  two  cases  : 

[24261],  when  653  does  not  vanish; 
[2427],    when  &53  vanishes. 

[24281].  The  defining  equation  of  this  case  can  be  reduced  to 


This   gives   a    quintuple  algebra  which  may  be  called  (bns),  its   multiplication 
table  being  t 


*This  algebra  is  mixed.  Namely,  if  &=j=l,  it  separates  on  substituting  t'j  —  (1  —  &)i  +  fc. 
k1  =  (l  —  b)  i  +  [a  (1  —  b)  +  1]  k  —  (1  —  &)  2m  ;  but  if  b  :=  1 ,  it  separates  on  substituting  i{  —  ai  —  (a2  +  a 
+  c)  fc  +  m  ,  &i  —  ai  +  g/c  +  ?/i .  [C.  S.  P.] 

t  Substitute  ij_  —  i  —  k ,  fcj  =  ak  +  ???, ,  and  the  algebra  separates.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 


103 


' 


j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

-I 

0 

0 

0 

al 

•  o 

0 

0 

0 

0 

j 

0 

j—al 

0 

9 

[2427].  The  defining  equation  of  this  case  is 


This   gives   a    quintuple  algebra  which  may  be  called  (bo5),    its  multiplication 
table  being  * 

(bo5)       i         j         It          I        m 


j 

0 

I 

0 

0 

0 

0 

0 

0 

0 

-I 

0 

0 

0 

al 

0 

0 

0 

0 

0 

j 

0 

—  al 

0 

<y 

1 

[243].  The  defining  equations  of  this  case  are 

0  —  ik  =  II  —  im , 


which  give 


*  Substitute  for  m ,  ai  +  m  ,  and  the  algebra  separates.     [C.  S.  P.] 


104 


PEIRCE  :    Linear  Associative  Algebra. 


There  are  two  cases  : 

[2431],  when  Id  =  I,  U  =  in  ,  mi  =  0  ; 
[2432],  when  Id  =  7,  H  =  mi  =  0. 

[2431].  The  defining  equations  of  this  case  are 

Id  =  1,     li  =•  m  ,     mi  =  0  , 
which  give 

l-j  =.  m  ,     Ij  =  mj  =  0  =  lie  =  ink  =  ?2  =  1m  =  ml 
0  —  H&  =  ild  =•  ikm  —  a3  =  r/.34  =  «35  , 


^'  =  yfem  =  csm  ,     0  =  l£m  =c3  =  Jem  , 
0  =  ¥  =  bm  +  dm  = 


There  are  two  cases  : 


[24312],   when  es  does  not  vanish  ; 
[24312],  when  e3  vanishes. 

[24312].  The  defining  equation  of  this  case  can  be  reduced  to 


This  gives  a  quintuple  algebra  which  may  be  called  (6/>5),    its    multiplication 
table  being  * 


*  The  structure  of  this  algebra  may  be  exhibited  by  putting    kl  =: 
—  —  a~*m ,  when  the  multiplication  table  becomes 

i        i         k         I        m 


=i  +  a   \j  —  a   'fc,   1^—j  —  a   V. 


i 

3 

0 

3 

0 

0 

3 

0 

0 

0 

° 

0 

k 

I 

m 

0 

0 

0 

I 

m 

0 

m 

0 

0 

m 

0 

0 

0 

0 

0 

In  relative  form,  i  =  B:C+  C:D,  j—B:D,  k  =  A:B  +  C:D,  l  =  A:C,  m—A\D.    [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 
(bp5)        i  j  ^  I  n 


105 


i 

j         o         o 

0 

0 

j 

0 

0 

0 

0 

0 

Jc 

I 

m 

-«*H- 

al-\-m 

am 

0 

I 

m            0 

0 

0 

0 

m 

0             0 

0 

0 

• 

0 

[24312].  The  defining  equation  of  this  case  is 


This    gives    a    quintuple  algebra  which  may  be  called  (635),  its  multiplication 
table  being* 

(bq5)     i         j         If          I         m 


i 

j 
Tf 
I 
m 

!].  The  definin 

J         0     :     ° 

0         0 

• 

000 

0         0 

I 

I         m        \     J 

am        0 

m         0          0 

0         0 

000 

0         0 

g  equations  of  this  case  are 
ki  =  l,     li  =  mi  =•  0  , 

*On  substituting    kl:=i  —  a~lk,  ll-=ij  —  a~ll, 
form  given  in  the  last  note.     [C.  S.  P.] 


^a"1?^,   this  algebra  reduces  to  (&p5).  in  the 


106 


PEIRCE  :    Linear  Associative  Algebra. 


which  give 


0  =  kj  =  Ij  •=.  mj  =  lk  =  1?  =  1m  =  itf  =  as  , 


0  =  ikm  =•  a35  =  kmi  =  c35  =  l£m  =  e35  =  imk  =.  a53  , 
ml  =  c^l  ,     0  =  msl  =  c53  =  ml  =  ml<?  =  e53  ; 


and  it  may  be  assumed  that 
which  gives 


1   =  m , 
=1  k3  =  km  =  mk  = 


There  is  then  a  quintuple  algebra  which  may  be  called  (br$),  its  multiplication 
being  * 

(br5)      i          j          k          1         m 


k 


m 


j 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

0 

, 

m 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[25].  The  denning  equations  of  this  case  are 

0  =  i2  =j2  =  1&  =  Z2  =  77i2  —  ij  -\-ji  ==.  ik  -\-  ki  =  il  -\-  li 
0  =jk  -\-  Itj  =jl  +  Ij  =jm  -f-  mj  =  Id  -\-  Ik  =  km  -f- 

and  it  may  be  assumed  that 

ij  =nk  =  —  ji  ,     il  =  m  =  —  li] 


=  im  +  mi  , 

=.  Im  +  ml  ; 


*In  relative  form.  i 
[C.  S.  P.] 


=  D:  E+  E  \F  ,   j=D:F,    k=.  A  :B  +  B  :  jP+  C:  E.    l  =  C:F,   m  =  A:F 


PEIRCE  :    Linear  Associative  Algebra.  107 

which  gives 

0  =  ik  •=.  lei  =  jk  =  kj  =  im  —  mi  —  km  =  mk  =  lm  =  ml , 
ijk  =  kl  •=.  b^k  -f-  dum  =  —  ilj  =  — mj  =jm , 
0  =  fm  =  dMjm  =  du  =  kP  =  b2Jd  =  684  =  M=  Ik  =jm  =  mj , 
0  — /27  =  a24/£  =  «24 , 


so  that  it  is  easy  to  see  that  there  is  no  pure  algebra  in  this  case. 

SEXTUPLE  ALGEBRA. 
There  are  two  cases  : 

[1],  when  there  is  an  idempotent  basis; 
[2],  when  the  algebra  is  nilpotent. 

[1],  The  defining  equation  of  this  case  is 

There  are  19  cases  : 

[I2],    when  all  the  other  units  but  i  are  in  the  first  group ; 

[12],   when/,  k,  I,  m  are  in  the  first  and  n  in  the  second  group  ; 

[13],   when/,  k  and  /  are  in  the  first  and  m  and  n  in  the  second  group ; 

[14],   when/,  k  and  /  are  in  the  first,  m  in  the  second  and  n  in  the  third  group; 

[15],   when/  and  k  are  in  the  first  and  /,  m  and  n  in  the  second  group ; 

[16],   when  /  and  k  are  in  the  first,  /  and  m  in  the  second  and  n  in  the  third 

group ; 
[17],  when/  and  k  are  in  the  first,  /  in  the  second,  m  in  the  third,  and  n  in  the 

fourth  group ; 

[18],   when/  is  in  the  first,  and  k,  I,  m  and  n  in  the  second  group ; 
[19],  when/  is  in  the  first,  k,  I  and  m  in  the  second,  and  n  in  the  third  group  ; 
[10'],  when  /  is  in  the  first,  k  and  /  in  the  second,  and  m  and  n  in  the  third  group  ; 
[11'],  when  /  is  in  the  first,  k  and  /  in  the  second,  m  in  the  third  and  n  in 

the  fourth  group  ; 
[12'],  when  /  is  in  the  first,  k  in  the  second,  I  in  the  third  and  m  and  n  in  the 

fourth  group ;    . 


108  PEIRCE  :  Linear  Associative  Algebra. 

[13'],  when/,  k,  I  ,  m  and  n  are  in  the  second  group  ; 

[14'],  when/,  k,  I  and  m  are  in  the  second  and  m  in  the  third  group  ; 

[15'],  when/,  k  and  I  are  in  the  second  and  m  and  n  are  in  the  third  group; 

[16'],  when  /,  k  and  /  are  in  the  second,  m  in  the  third,  and  n  in  the  fourth 

group  ; 
[17'],  when/  and  k  are  in  the  second,  /  and  m  in  the  third,  and  n  in  the  fourth 

group  ; 
[18'],  when/  and  k  are  in  the  second,  I  in  the  third,  and  m  and  n  in  the  fourth 

group  ; 
[19'],  when/  is  in  the  second,  k  in  the  third,  and  /,   m  and  n  in  the  fourth  group. 

[I2]  ;  The  defining  equations  of  this  case  are 
ij  =ji  =-j  ,     ik  =  ki  =  k  ,     il  =  li  =  I  ,     im  =  mi  =  m  ,     in  =  ni  =•  n  . 

and  the  54  algebras  of  this  case  deduced  from  (5-5)  to  (br^)  may  be  called  (a6)  to 

(»,).* 

[12].  The  defining  equations  of  this  case  are 

ij  =ji  —  /  ,     ik  —  ki  =•  k  ,     il  =  li  =  I  ,     im  =•  mi  =•  m  ,     in  =  n  ,     ni  —  0  , 

which  give 

0  =jn  —  /a/  =  kn  =  nk  =  ln  =  nl  =  mn  =  nm  ==.  nz, 

so  that  there  is  no  pure  algebra  in  this  case. 

[13].  The  defining  equations  of  this  case  are 

ij  =ji  =  j,     ik  =.  ki  =  k  ,     il  =  li  —  I  ,     im  =  m  ,     in  =  n  ,     7m  =  m  =  0  . 

There  are  four  cases,  which  correspond  to  relations  between  the  units  of  the 
first  group  similar  to  those  of  the  quadruple  algebras  (a4)  ,  (64)  ,  (c4)  or  (c£4)  . 
[131].  The  defining  equations  of  this  case  are 


and,  in  the  result,  we  obtain 

jm  =  n  ,     jn—  km  =  kn  =  Im  =.  In  =  0  . 

*  The  multiplication  tables  of  these  algebras,  formed  from  the  nilpotent  quintuple  algebras,  in  the 
same  manner  in  which  the  first  class  of  quintuple  algebras  are  formed  from  the  nilpotent  quadruple 
algebras,  have  been  omitted.  [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra.. 


109 


This  gives  a  sextuple  algebra  which  may  be  called  (bc6),  of  which  the  multipli 
cation  table  is  * 

(bce)      i          j          Tf          I          m         n 


m 


i          j 

If       I 

j 

m         n 

j          * 

I         0 

, 

n         0 

Jc          I 

0         0 

0         0 

I         0 

0         0 

• 

0         0 

0         0 

i 

0         0 

0         0 

0         0 

0         0 

0         0 

[132].  The  defining  equations  of  this  case  are 
z  =  Jc  =  l\     Ij—ak,    jk=jl=fy'= 


which  give 

There  are  two  cases : 


km  =  kn  —  0 . 


[1321],  when  <?24  does  not  vanish  ; 
[1322],    when  e24  vanishes. 

[1321].  The  defining  equation  of  this  case  can  be  reduced  to 

jn  =  m , 
which  gives 

0  =jm  =  Im  . 

This  gives  a  sextuple  algebra  which  may  be  called  (&c£6),  of  which  the  multipli 
cation  table  is  f 


*  In  relative  form,  i  —  A:  A  +  B  :B  +  C  :  (7+  D:D,j  =  A:B+B:  C+  C:D.  k  =  A:  C  +  B:D. 
l  —  A:D,m  —  B:E,n  —  A:E.  [C.  S.  P.] 

t  This  algebra  is  distinguishable  into  two.  in  the  same  manner  as  (c3).  Namely,  if  a=  ±  2  .  on  sub 
stituting  l^—l±j.  we  have  Z2  =  0  .  jl  —  k.  lj=  —  k,  and  the  multiplication  table  is  otherwise  un 
changed.  Otherwise,  on  substituting  j\  —  Z-f  cj,  ll=zk  +  c~lj  .  where  2e  —  —  adr^/a2  —  4.  we  have 
J2=l2  =  0.jl=(l  —  c2)k.  lj=(\  —  c--)k,jn=(b-\-c)k.  In  —  (b-f  c-1)  k.  and  otherwise  the  multi 
plication  table  is  unchanged.  The  following  is  a  relative  form  for  the  first  variety:  i=.A  :A  +  B  :B 


For  the  second  variety,  i  =  A  :  A  +  B  :  B+  C  :  C'-f  D  :  D,  j  =.  A  :  B+  (1  —  c3)  C:D.  k  =  A  :D,  l  =  A:C 
+  (1  —  c~*)B:D.  m  =  A:E.  n=  (b  +  c)  B  :E+  (b  +  c-1)  C  :  E.     [C.  S.  P.] 


MO 


PEIRCE  :    Linear  Associative  Algebra. 
(bd6)      i          j          k          I         m 


i 

j 
* 
I 
m 
n 

i 

1 
j 

& 

I 

m 

n 

j 

Tf 

0 

0 

0 

m 

Tt 

0 

0 

0 

0 

0 

I 

ok 

0 

A- 

0 

bm 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[1322].  The  defining  equation  of  this  case  is 

jn  =  0  > 

and  there  is  no  pure  algebra  in  this  case. 

[132].  The  defining  equations  of  this  case  are 


which  give 


km  =  kn  =  0 


There  is  a  sextuple  algebra  in  this  case  which  may  be  called  (&ee),  of  which  the 
multiplication  table  is  * 


*This  algebra  may  be  a  little  simplified  by  substituting  j—lforj.  In  relative  form,  i  —  A:  A 
+  B:B+  C:C+D:Z>,  j  =  A:D+B:C,  k  —  A:C,  l  =  A:B,  m  =  A:E,  n  —  bB:E+aD:E. 
[C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 
(be6)      i          j          k          I         m         n 


111 


i 

i 

j 

If 

/ 

m 

n 

j 

j 

If 

0 

0 

0 

am 

Jc 

If, 

0 

0 

0 

0 

0 

I 

I 

Tf 

0 

0 

0 

bm 

m 

0 

0 

0 

0 

0 

0 

n 

0 

0 

0 

0 

0 

0 

[134],  The  defining  equations  of  this  case  are 

jk  =  —kj=  I,    /=  tf=jJc.  =  kj=kl=lk=  /2  =  0. 

There  is  a  sextuple  algebra  in  this  case  which  may  be  called  (fe/e),  of  which  the 
multiplication  table  is  * 


m          n 


i 

i 

j 

Tf 

I 

m 

n 

j 

j 

0 

0 

Tf 

0 

m 

Jc 

k 

0 

0 

0 

0 

0 

I 

I 

—  Jc 

0 

0 

0 

am 

m 

0 

0 

0 

0 

0 

0 

n 

0 

0 

0 

0 

0 

0 

*  In  relative  form,  t  =  4  :4  +  J5:B+  C:C+  D:D,  j=A:B—C:D,  k  —  A:D,  l  =  A:C+B:D, 
=  A:E,  n  =  B  :E+aC:E .    [C.  S.  P.] 


112  PEIRCE  :    Linear  Associative  Algebra. 

[14].  The  defining  equations  of  this  case  are 

ij  —  ji  —  j ,     ik  =  M  =  k  ,     il  =  li  =  I ,     im .  =.  m  ,     ni  =  n ,     mi  =  in  =  0 , 
which  give 
0  =jm  =jn  =  km  =  kn  =  lm  =  ln  =  mj  =  nj  =  mk  =  nk  =  ml=.nl  =  mz=nm  =nz 

There  are  four  cases  denned  as  in  [13]. 

[141].  The  defining  equations  of  this  case  are 


which  give 


mn  = 


There  is  a  sextuple  algebra  which  may  be  called 
cation  table  is* 

i  Jf          I 


m 


n 


of  which  the  multipli- 


m         n 


i 

j 

k 

' 

m 

0 

j 

k 

I 

0 

0 

0 

Jf 

I 

0 

0 

0 

0 

I 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

I 

n 

0 

0 

0 

0 

0 

[142].   The    denning   equations    of  this    case    are    the    same    as  in   [132], 

which  give 

mn  =• 


*In  relative  form,  i  =  A:A  +  B:B+C:C+D:D,j= A  :B+ B  :C+ C  :D .  k  =  A:C+ B  :D 
A:D.m  —  A:E,n  =  E:D.     [C.  S.  P.] 


PEIRCE  :    Linear  Associative  Algebra. 


113 


There  is  a  sextuple  algebra  which  may  be  called  (bh9),   of  which  the  multipli 
cation  table  is  * 

i         j         k          I         m         n 


k 


m 


n 


i 

. 
.1 

k 

I 

m 

0 

j 

Tf 

0 

0 

0 

0 

~k 

0 

0 

0 

0 

0 

I 

ok 

0 

k 

0 

0 

0 

0 

0 

0 

0 

k 

n 

0 

0 

0 

0 

0 

[143].  The  denning  equations  of  this  case  are  the  same  as  in  [132].  There 
is  a  sextuple  algebra  which  may  be  called  (U,\  of  which  the  multiplication 
table  isf 

(fo'e)      i          j          If          I         m         n 


i 

j 

* 

/ 

m 

1 
0 

j 

k 

0 

0 

0 

0 

A- 

0 

0 

0 

0 

0 

I 

k 

0 

0 

0 

0 

0 

0 

0 

0 

0 

* 

n 

0 

0 

0 

0 

0 

*  This  algebra  has  two  varieties,  analogous  to  those  of  (c3 ).     The  first  is,  in  relative  form,  i  =  A-A 

+  B:B+C:C+D:D,j=A:B  +  B:C+A:D,lc=A:C.l=-A:B  +  D:C,m  =  A:E.n  =  E:.C. 
m  relative  form  is  the  same  except  that./  —  A:B-\-  6-'Z> :  C,  l  =  A:D  —  bB  •  C      [CSP] 

g?ra  maj  be  Sligh%  simPlified  by  PuttiQg  •/- 1  forj.     Then,  in  relative  form.  i=A'A 
:C.  j  =  B  :  C .  k-A:C,  l-A:B.  m  =  A:D.  n  =  D:C.     [C.  S.  P.] 


114 


PEIRCE  :    Linear  Associative  Algebra. 


[143].  The  defining  equations  of  this  case  are  the  same  as  in  [134].  There 
is  a  sextuple  algebra  which  may  be  called  (6/6),  of  which  the  multiplication 
table  is  * 


m         n 


i 

j 

It       I 

m 

0 

j 

0 

0         A- 

0 

0 

k 

0 

0         0 

o 

0 

^ 

fc 

0         () 

0 

0 

0 

0 

0         0 

1 

0 

If 

» 

0 

0         0 

0 

0 

[15].   The  defining  equations  of  this  case  are 

ij  =  ji  =  j ,     ik  =  ki  =  k,     il=J,     im  =  m  ,     wi  =  w ,     /i  =  wii  =  ni  =  0  , 
which  give 

/  =  k,     0  =jk  =  kj  =  tf  =  /;'  =  //«'  =  /*  =  lm  =  1n=  mj  —  mk  =  ml  =  in1 

•=.  mn  =  nj  =•  nk  =  nl  =  nm  =  ?i2. 

There  is  a  sextuple  algebra  which  may  be  called  (/>7,-6),  of  which  the  multipli 
cation  table  isf 


*t  =  A\A  +  B:B+  C\C+D:D,   j—A:B  —  C:D,    k  —  A:D.    l  —  A: 
E:D.     [C.  S.  P.] 

tin  relative  form,  t  =  A  :  A  +  B  :  B+  C  :  C,  j=A:B  +  B:C,   k  —  A  :  C. 
A:D.     [C.  S.  P.] 


,    m  —  A:E. 
=  C  :  D,  m  =  B  :D. 


PEIRCE  :    Linear  Associative  Algebra. 
(bk6)      i          j          k          I         m         n 


115 


k 


tn 


n 


i 

j 

k 

I 

m 

ft 

j 

k 

0 

m 

n 

0 

k 

0 

0 

it 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

[16].  The  defining  equations  of  this  case  are 

ij  =ji  —  J  ,     ik  =  ki  =  k ,     ^7  =  / ,     im  =  m  ,     ni  —  n ,     U  =  mi  =.  in  =  0 , 
which  give 

y*  =  k ,     0  =-jk  =jn  =  kj  =  k*  =  kn  =  Ij  =  lk  =  I*  =  lm  =  mj  =  mk  =.  nl  =  m 

=.  nj  =•  nk  =  nl  =•  nm  =  ri*. 

There  is  a  sextuple  algebra  which  may  be  called  (&/6),  of  which  the  multipli 
cation  table  is  * 

(bl6)      i          j          k          I         m         n 


k 


n 


i 

j 

K 

I 

m 

0 

j 

k 

0 

m 

0 

0 

k 

0 

0 

0 

0 

0 

0 

0 

o 

0 

0 

k 

0 

0 

0 

0 

0 

0 

n 

0 

0 

0 

0 

0 

*In   relative   form.   i  =  A:  A  +  B  :  B  +  C :  C,   j—A:B  +  B:C.    k  —  A:C,    l  =  B: 
=  A:D,  n  —  E  :C.     [C.  S.  P.] 


116  PEIRCE  :   Linear  Associative  Algebra. 

[17].  The  defining  equations  of  this  case  are 
ij  =ji  ==j ,     ik  —  Id  =  k ,     il  =.  I ,     mi  :=  m  ,     /i  =  iwi  =  in  =  m  =  0 . 

There  is  no  pure  algebra  in  this  case. 

[18].  The  defining  equations  of  this  case  are 

ij  =  ji  =  j,     ik  =  k ,  il  =  I ,     im  =  m  ,     in  =  n  ,     ki  =  li=  mi  =  ni  =  0  . 

There  is  no  pure  algebra  in  this  case. 

[19].  The  defining  equations  of  this  case  are 

ij  =ji  =  j,     ik  =•  k ,     il  =  I ,     im  =.  m  ,     ni  =  n ,     in  =  kl  =  li  =  ni  =  n . 

There  is  no  pure  algebra  in  this  case. 

[10'].  The  defining  equations  of  this  case  are 

ij  =-ji  =  j,     ik  —  lt,     il  =  I ,     mi  =  m ,     ni  =  n ,     i/n  =  ^w  =  &£  =  li  =  0 . 

There  is  no  pure  algebra  in  this  case. 

[11'].  The  defining  equations  of  this  case  are 

ij  =-ji  =-j ,     ik  =  k,     il  =.  I ,     mi  =  m  ,     im  =  Zt'  =  m  =  m  =  0. 

There  is  no  pure  algebra  in  this  case. 

[12'].  The  defining  equations  of  this  case  are 

ij  =ji  —j,     ik  =  k,     li  —  / ,     il  =.  im  =  in  •=.ld'=-  mi  =•  ni  =  0  . 

There  is  no  pure  algebra  in  this  case. 

[13'].  The  defining  equations  of  this  case  are 

ij  —j ,     ik  ==•  k ,     il  =•  I ,     im  =  m ,     in  =  n  ,     ji  =  ki  —  li  =  mi  •=  ni  =  0 . 

There  is  no  pure  algebra  in  this  case. 

[14'].  The  defining  equations  of  this  case  are 

ij  =j ,     ik  •=•  k ,     il  =  I ,     im  =•  m  ,     ni  =  n  ,     /&  =  /M  =  li  =  mi  =  in  =  Q  . 

There  is  no  pure  algebra  in  this  case. 

[15'].  The  defining  equations  of  this  case  are 

ij  =  j ,     ik  =•  k  ,     il  =  I,     mi  •=.  m  ,     ni  •=•  n  ,     im  =  in  =  ji  =  ki  =  ?i  =r  0  . 
There  is  no  pure  algebra  in  this  case. 


PBIRCE  :   Linear  Associative  Algebra.  117 

[16'].  The  defining  equations  of  this  case  are 

*}'  =y '     ik  =  k,     il  =  I ,     mi  =  m  ,     im  =  in  =  jk  =  kl  =  li  =  ni  =  0 . 
There  is  no  pure  algebra  in  this  case. 

[17'].  The  defining  equations  of  this  case  are 

ij  =y ,     ik  =  k ,      U=l,     mi  =  m  ,     ?7  =  iwi  =  m  =  ji  =.  ki  =•  ni  =  0  . 
There  is  no  pure  algebra  in  this  case. 

[18'].  The  defining  equations  of  this  case  are 

if  =y  ,     ik  =  k ,     li  —  l,    ji  =  ki  =  il  =  im  =  in  =  mi  =  m  =  0  . 
There  are  six  cases  : 

[18'1],  when  m*  =  m  ,  mn  =  n  ,  nm  =  0  , 
[18'2],  when  m*  ==  m  ,    mn  =  0  ,  nm  =  n , 
[18'3],  when  w2  =  n  ,     mn  =  nm  =  0  ,  ri*  =  w  , 
[18' 4],  when  mz  =  m  ,  mn  =  T^TZ  =  yi2 1=  0  , 
[18'5],  when  m2  =  n  ,     m3  =  0  , 
[18'6],  when  w2  =  ^2  =  0. 

[18'1].  The  defining  equations  of  this  case  are 

mz  =.  m  ,     mn  =  n  ,     nm  —  0  . 
There  are  two  cases  : 

[18'12],    when»iZ  =  0; 
[1812],  when  ml=l. 
[18'12].  The  defining  equation  of  this  case  is 

mf=0. 
There  is  no  pure  algebra  in  this  case. 

[18'12].  The  defining  equation  of  this  case  is 

ml— I. 
There  are  two  cases  : 

[18'121],  when  >!=./; 
[18122],   whenym  =  0. 

[18'121].  The  defining  equation  of  this  case  is 

jm  =j. 

There  is  a  sextuple  algebra  which  may  be  called  (6m6),  of  which  the  multipli 
cation  table  is  * 

*In  relative  form,  i  =  A:A,j=A:B,  k  =  A:C,  l=.B:A,  m  =  B:B,  n  =  B:C.    [C.  S.  P.] 


118 


PEIRCE  :    Linear  Associative  Algebra. 
(bm6)      i          j          k          I          m         n 


m 


n 


i 

> 

0 

0 

0 

0 

0 

0 

$ 

J 

* 

0 

0 

0 

o 

o 

0 

I 

m 

n 

0 

0 

0 

0 

0 

0 

I 

m 

w 

0 

0 

0 

0 

0 

0 

[18'122].  The  defining  equation  of  this  case  is 

jm  =.  0 . 

There  is  no  pure  algebra  in  this  case. 

[18'2].  The  defining  equations  of  this  case  are 

m?  =.  m  ,     mn  =  0  ,     nm  =•  n . 
There  are  two  cases : 

[18'21],  when  ml  =  I; 

[18'23],   whenml=Q. 

[18'21].  The  defining  equation  of  this  case  is 

ml  —  I . 
There  are  two  cases : 

[18'2P],   whenyw=y; 

[18'212],  whenym=0. 

[18'212].  The  defining  equation  of  this  case  is 

jm=j. 

There  is  no  pure  algebra  in  this  case. 

[18'212].  The  defining  equation  of  this  case  is 

jm  =•  0 . 
There  is  no  pure  algebra  in  this  case. 


PEIRCE  :   Linear   Associative  Algebra.  119 

[18'22].  The  defining  equation  of  this  case  is 

ml  =  0. 

There  is  no  pure  algebra  in  this  case. 

[18'3].  The  defining  equations  of  this  case  are 

m2  =  m  ,     mn  =  nm  =.  0  ,     n2  =  n . 

There  is  no  pure  algebra  in  this  case. 

[18' 4],  The  defining  equations  of  this  case  are 

m3  =  m ,     mn  =  nm  =  nz  =  0 . 
There  are  two  cases  : 

[18'41],  when/wi  =j; 

[18'42],  when/m  =  0. 

[18'41].  The  defining  equation  of  this  case  is 

jm  — ./. 

There  is  no  pure  algebra  in  this  case. 

[18'42].  The  defining  equation  of  this  case  is 

jm  •=.  0 . 

There  is  no  pure  algebra  in  this  case. 

[18'5].  The  defining  equations  of  this  case  are 

m2  —  n  ,     m3  —  0  . 

There  is  no  pure  algebra  in  this  case. 

[18'6].  The  defining  equations  of  this  case  are 

m2  —  n*  =  0  . 

There  is  no  pure  algebra  in  this  case. 

[19'].  The  defining  equations  of  this  case  are 

ij  =j ,     ki  •=.  k ,    ji  =  ik  =  il  =  im  =:  in  =  li  =  mi  =  ni  =  0  . 

There  is  no  pure  algebra  in  this  case. 

[2].  The  algebras  belonging  to  this  case  are  not  investigated,  because  it  is 
evident  from  §  69  that  they  are  rarely  of  use  unless  combined  with  an  idempo- 
tent  basis,  so  as  to  give  septuple  algebras. 

NATURAL  CLASSIFICATION. 

There  are  many  cases  of  these  algebras  which  may  obviously  be  combined 
into  natural  classes,  but  the  consideration  of  this  portion  of  the  subject  will  be 
reserved  to  subsequent  researches. 


120  PEIRCE  :   Linear  Associative  Algebra. 


ADDENDA. 
I. 

On  the    Uses  and   Transformations  of  Linear  Algebra. 
BY  BENJAMIN  PEIRCE. 

[Presented  to  the  American  Academy  of  Arts  and  Sciences,  May  11,  1875.] 

Some  definite  interpretation  of  a  linear  algebra  would,  at  first  sight,  appear 
indispensable  to  its  successful  application.  But  on  the  contrary,  it  is  a  singular 
fact,  and  one  quite  consonant  with  the  principles  of  sound  logic,  that  its  first  and 
general  use  is  mostly  to  be  expected  from  its  want  of  significance.  The  interpre 
tation  is  a  trammel  to  the  use.  Symbols  are  essential  to  comprehensive  argument. 
The  familiar  proposition  that  all  A  is  B,  and  all  B  is  G,  and  therefore  all  A  is  C, 
is  contracted  in  its  domain  by  the  substitution  of  significant  words  for  the 
symbolic  letters.  The  A,  B,  and  G,  are  subject  to  no  limitation  for  the  purposes 
and  validity  of  the  proposition ;  they  may  represent  not  merely  the  actual,  but 
also  the  ideal,  the  impossible  as  well  as  the  possible.  In  Algebra,  likewise,  the 
letters  are  symbols  which,  passed  through  a  machinery  of  argument  in  accord 
ance  with  given  laws,  are  developed  into  symbolic  results  under  the  name  of 
formulas.  When  the  formulas  admit  of  intelligible  interpretation,  they  are 
accessions  to  knowledge ;  but  independently  of  their  interpretation  they  are 
invaluable  as  symbolical  expressions  of  thought.  But  the  most  noted  instance 
is  the  symbol  called  the  impossible  or  imaginary,  known  also  as  the  square  root 
of  minus  one,  and  which,  from  a  shadow  of  meaning  attached  to  it,  may  be 
more  definitely  distinguished  as  the  symbol  of  semi-inversion.  This  symbol  is 
restricted  to  a  precise  signification  as  the  representative  of  perpendicularity  in 
quaternions,  and  this  wonderful  algebra  of  space  is  intimately  dependent  upon 
the  special  use  of  the  symbol  for  its  symmetry,  elegance,  and  power.  The 
immortal  author  of  quaternions  has  shown  that  there  are  other  significations 
which  may  attach  to  the  symbol  in  other  cases.  But  the  strongest  use  of  the 
symbol  is  to  be  found  in  its  magical  power  of  doubling  the  actual  universe,  and 


PEIRCE  :    Linear  Associative  Algebra.  121 

placing  by  its  side  an  ideal  universe,  its  exact  counterpart,  with  which  it  can  be 
compared  and  contrasted,  and,  by  means  of  curiously  connecting  fibres,  form 
with  it  an  organic  whole,  from  which  modern  analysis  has  developed  her 
surpassing  geometry.  The  letters  or  units  of  the  linear  algebras,  or  to  use  the 
better  term  proposed  by  Mr.  Charles  S.  Peirce,  the  vids  of  these  algebras,  are 
fitted  to  perform  a  similar  function  each  in  its  peculiar  way.  This  is  their 
primitive -and  perhaps  will  always  be  their  principal  use.  It  does  not  exclude 
the  possibility  of  some  special  modes  of  interpretation,  but,  on  the  contrary,  a 
higher  philosophy,  which  believes  in  the  capacity  of  the  material  universe  for 
all  expressions  of  human  thought,  will  find,  in  the  utility  of  the  vids,  an  indica 
tion  of  their  probable  reality  of  interpretation.  Doctor  Hermann  Hankel's 
alternate  numbers,  with  Professor  Clifford's  applications  to  determinants,  are  a 
curious  and  interesting  example  of  the  possible  advantage  to  be  obtained  from 
the  new  algebras.  Doctor  Spottiswoode  in  his  fine,  generous,  and  complete 
analysis  of  my  own  treatise  before  the  London  Mathematical  Society  in  Novem 
ber  of  1872,  has  regarded  these  numbers  as  quite  different  from  the  algebras 
discussed  in  my  treatise,  because  they  are  neither  linear  nor  limited.  But  there 
is  no  difficulty  in  reducing  them  to  a  linear  form,  and,  indeed,  my  algebra  (<?3)  is 
the  simplest  case  of  Hankel's  alternate  numbers  ;  and  in  any  other  case,  in  which 
n  is  the  number  of  the  Hankel  elements  employed,  the  complete  number  of  vids 
of  the  corresponding  linear  algebra  is  (2n  —  1 .  The  limited  character  of  the 
algebras  which  I  have  investigated  may  be  regarded  as  an  accident  of  the  mode 
of  discussion.  There  is,  however,  a  large  number  of  unlimited  algebras 
suggested  by  the  investigations,  and  Hankel's  numbers  themselves  would  have 
been  a  natural  generalization  from  the  proposition  of  §  65  of  my  algebra. * 
Another  class  of  unlimited  algebras,  which  would  readily  occur  from  the 
inspection  of  those  which  are  given,  is  that  in  which  all  the  powers  of  a  vid  are 
adopted  as  independent  vids,  and  the  highest  power  may  either  be  zero,  or  unity, 
or  the  vid  itself,  and  the  zero  power  of  the  fundamental  vid,  i.  e.  unity  itself, 
may  also  be  retained  as  a  vid.  But  I  desire  to  draw  especial  attention  to  that 
class,  which  is  also  unlimited,  and  for  which,  when  it  was  laid  before  the  math 
ematical  society  of  London  in  January  of  1870,  Professor  Clifford  proposed  the 
appropriate  name  of  quadrates. 

*  This  remark  is  not  intended  as  a  foundation  for  a  claim  upon  the  Hankel  numbers,  which  were 
published  in  1867.  three  years  prior  to  the  publication  of  my  own  treatise. — B.  P.  [They  were  given 
much  earlier  under  the  name  of  clefs  by  Cauchy,  and  (substantially)  at  a  still  earlier  date  by  Grassmann. 
— C.  S.  P.] 


122  PEIRCE:   Linear  Associative  Algebra. 

Quadrates. 

The  best  definition  of  quadrates  is  that  proposed  by  Mr.  Charles  S.  Peirce. 
If  the  letters  A,  B,  C,  etc.,  represent  absolute  quantities,  differing  in  quality, 
the  vids  may  represent  the  relations  of  these  quantities,  and  may  be  written  in 
the  form 

(A:A)(A:B)(A:O)   .   .   .   (B  :  A)  (B  :  B)  .   .   .   (C:A),   etc. 

subject  to  the  equations 

(A  :B)(B:G)  =  (A:  C) 

(A:B)(C:D)  =  Q. 

In  other  words,  every  product  vanishes,  in  which  the  second  letter  of  the  multi 
plier  differs  from  the  first  letter  of  the  multiplicand ;  and  when  these  two  letters 
are  identical,  both  are  omitted,  and  the  product  is  the  vid  which  is  compounded 
of  the  remaining  letters,  which  retain  their  relative  position. 

Mr.  Peirce  has  shown  by  a  simple  logical  argument  that  the  quadrate  is  the 
legitimate  form  of  a  complete  linear  algebra,  and  that  all  the  forms  of  the 
algebras  given  by  me  must  be  imperfect  quadrates,  and  has  confirmed  this 
conclusion  by  actual  investigation  and  reduction.  His  investigations  do  not 
however  dispense  with  the  analysis  by  which  the  independent  forms  have 
been  deduced  in  my  treatise,  though  they  seem  to  throw  much  light  upon  their 
probable  use. 

Unity. 

The  sum  of  the  vids  (A  :  A),  (B  :B),  (C :  G),  etc.,  extended  so  as  to  include 
all  the  letters  which  represent  absolute  quantities  in  a  given  algebra,  whether  it 
be  a  complete  or  an  incomplete  quadrate,  has  the  peculiar  character  of  being 
idempotent,  and  of  leaving  any  factor  unchanged  with  which  it  is  combined  as 
multiplier  or  multiplicand.  This  is  the  distinguishing  property  of  unity,  so  that 
this  combination  of  the  vids  can  be  regarded  as  unity,  and  may  be  introduced 
as  such  and  called  the  vid  of  unity.  There  is'  no  other  combination  which 
possesses  this  property. 

But  any  one  of  the  vids  (A:  A),  (B  :  B),  etc.,  or  the  sum  of  any  of  these 
vids  is  idempotent.  There  are  many  other  idempotent  combinations,  such  as 

(A:A)  +  x(A:B),     y  (A  :  B)  +  (B  :  B), 
^(A:A)-\-^(A:B)  +  ^(B:A)  +  ^(B:B), 

which  may  deserve  consideration  in  making  transformations  of  an  algebra 
preparatory  to  its  application. 


PEIRCE  :    Linear  Associative  Algebra,.  123 

Inversion. 

A  vid  which  differs  from  unity,  but  of  which  the  square  is  equal  to  unity, 
may  be  called  a  vid  of  inversion.  For  such  a  vid  when  applied  to  some  other 
combination  transforms  it ;  but,  whatever  the  transformation,  a  repetition  of  the 
application  restores  the  combination  to  its  primitive  form.  A  very  general  form 
of  a  vid  of  inversion  is 

(A  :  A)  ±  (B  :  B)  ±  (G :  C)  ±  etc., 

in  which  each  doubtful  sign  corresponds  to  two  cases,  except  that  at  least  one  of 
the  signs  must  be  negative.  The  negative  of  unity  might  also  be  regarded  as  a 
symbol  of  inversion,  but  cannot  take  the  place  of  an  independent  vid.  Besides 
the  above  vids  of  inversion,  others  may  be  formed  by  adding  to  either  of  them 
a  vid  consisting  of  two  different  letters,  which  correspond  to  two  of  the  one- 
lettered  vids  of  different  signs ;  and  this  additional  vid  may  have  any  numerical 
coefficient  whatever.  Thus 

(A:A)  +  (B:B)  —  (C:C)  +  x(A:  C)  +  y  (B  :  C) 

is  a  vid  of  inversion. 

The  new  vid  which  Professor  Clifford  has  introduced  into  his  biquaternions 
is  a  vid  of  inversion. 

Sem  i- Inversion. 

A  vid  of  which  the  square  is  a  vid  of  inversion,  is  a  vid  of  semi-inversion. 
A  very  general  form  of  a  vid  of  semi-inversion  is 

(A:A)±  (B:B)±J(C:  C)  ±  etc. 

in  which  one  or  more  of  the  terms  (A:  A),  (B  :  B),  etc.,  have  J  for  a  coeffi 
cient.     The  combination 

(A:A)±  J(B  :  B)  +  x(A  :  B)  +  etc. 

is  also  a  vid  of  semi-inversion.     With  the  exception  of  unity,  all  the  vids  of 
Hamilton's  quaternions  are  vids  of  semi-inversion. 

The  Use  of  Commutative  Algebras. 

Commutative    algebras   are    especially   applicable    to    the    integration    of 
differential  equations  of  the  first  degree  with  constant  coefficients.     If  i,  j,  k, 


124 


PEIRCE  :    Linear  Associative  Algebra. 


etc.,  are  the  vids  of  such  an  algebra,  while  x,  y,  z,  etc.,  are  independent 
variables,  it  is  easy  to  show  that  a  solution  may  have  the  form  F  (xi  +  yj  -\-  zk 
-fete.),  in  which  .Pis  an  arbitrary  function,  and  i,  ./,  k,  etc.,  are  connected  by 
some  simple  equation.  This  solution  can  be  developed  into  the  form 

F(xi  +  yj  +  zk  +  etc.)  =  Mi  +  Nj  +  Pk  +  etc. 

in  which  M,  N,  P,  etc.,  will  be  functions  of  x,  y,  z,  etc.,  and  each  of  them  is  a 
solution  of  the  given  equation.  Thus  in  the  case  of  Laplace's  equation  for  the 
potential  of  attracting  masses,  the  vids  must  satisfy  the  equation 

+#  =  0. 


The  algebra  («3)  of  which  the  multiplication  table  is 

i          j         k 


i 

j 

n 

j 

* 

0 

k 

0 

0 

may  be  used  for  this  case.  Combinations  il7  j\,  ^  of  these  vids  can  be  found 
which  satisfy  the  equation 

*!+/!  +  *!  =  o, 

and  if  the  functional  solution 

F(vii+yji  +  *ki) 

is  developed  into  the  form  of  the  original  vids 

M+  Nj+Pk, 

M,  N,  and  P  will  be  independent  solutions,  of  such  a  kind  that  the  surfaces  for 
which  N  and  P  are  constant  will  be  perpendicular  to  that  for  which  M  is 
constant,  which  is  of  great  importance  in  the  problems  of  electricity. 

The   Use  of  Mixed  Algebras. 

It  is  quite  important  to  know  the  various  kinds  of  pure  algebra  in  making 
a  selection  for  special  use,  but  mixed  algebras  can  also  be  used  with  advantage 


PEIRCE:    Linear  Associative  Algebra.  125 

in  certain  cases.  Thus,  in  Professor  Clifford's  biquaternions,  of  which  he  has 
demonstrated  the  great  value,  other  vids  can  be  substituted  for  unity  and  his 
new  vid,  namely  their  half  sum  and  half  difference,  and  each  of  the  original 
vids  of  the  quaternions  can  be  multiplied  by  these,  giving  us  two  sets  of  vids, 
each  of  which  will  constitute  an  independent  quadruple  algebra  of  the  same 
form  with  quaternions.  Thus  if  i,j,  k,  are  the  primitive  quaternion  vids  and 
ID  the  new  vid,  let 

tt!  =  iz  (1  +  w)  .  a8  =  2  (1  —  w). 

i^  rz:  a-ii  .  4  =  azi  . 


Then  since 

ai  —  ai  •  al  = 

— 


^  0  = 
=  0  = 

in  which  Ml  denotes  any  combination  of  the  vids  of  the  first  algebra,  and  Nz  any 
combination  of  those  of  the  second  algebra.  It  may  perhaps  be  claimed  that 
these  algebras  are  not  independent,  because  the  sum  of  the  vids  ax  and  a2  is 
absolute  unity.  This,  however,  should  be  regarded  as  a  fact  of  interpretation 
which  is  not  apparent  in  the  defining  equations  of  the  algebras. 

II. 

On  the  Relative  Forms  of  the  Algebras. 
BY  C.  S.  PEIRCE. 

Griven  an  associative  algebra  whose  letters  are  i,  /,./<",  /,  etc.,  and  whose 

multiplication  table  is 

i*  =•  aui  +  buj  -\-  cnk  +  etc.* 

ij  =  ai&  +  &12/  +  clzk  +  etc. 
ji  =  a^i  +  b2lj  +  c^k  +  etc., 
etc.,  etc. 

I  proceed  to  explain  what  I  call  the  relative  form  of  this  algebra. 


I  have  used  al  l  ,  etc.,  in  place  of  the  al  ,  etc..  used  by  my  father  in  his  text. 


126  PEIRCE  :    Linear  Associative  Algebra. 

Let  us  assume  a  number  of  new  units,  A,  /,  J,  K,  L,  etc.,  one  more  in 
number  than  the  letters  of  the  algebra,  and  every  one  except  the  first,  A , 
corresponding  to  a  particular  letter  of  the  algebra.  These  new  units  are  sus 
ceptible  of  being  multiplied  by  numerical  coefficients  and  of  being  added 
together ;  *  but  they  cannot  be  multiplied  together,  and  hence  are  called  non- 
relative  units. 

Next,  let  us  assume  a  number  of  operations  each  denoted  by  bracketing 
together  two  non-relative  units  separated  by  a  colon.  These  operations,  equal  in 
number  to  the  square  of  the  number  of  non-relative  units,  may  be  arranged  as 
follows : 

(A  :A)     (A:  /)     (A  :J)     (A:  K\  etc. 

(I:  A)       (1:1)      (I:J)      (I :  K),  etc. 

(J:A)      (/:/)      (J:J)      (J :  K\  etc. 

Any  one  of  these  operations  performed  upon  a  polynomial  in  non-relative  units, 
of  which  one  term  is  a  numerical  multiple  of  the  letter  folio  wing  the  colon,  gives  the 
same  multiple  of  the  letter  preceding  the  colon.  Thus,  (I:J)  («/+  bJ+  cK)  =  bI.-f 
These  operations  are  also  taken  to  be  susceptible  of  associative  combination. 
Hence  (/ :  J)  (J:  K)  =  (/  :  K)  •  for  (J :  K)  K  =  J  and  (I:J)J=  I,  so  that 
(I:J)(J:K)K=I.  And  (I:J)(K:L)  =  Q-,  for  (K:  L)  L  =  K  and  (/:  J)  K 
=  (/:/)  (Q.J+K)  =  0./=  0.  We  further  assume  the  application  of  the 
distributive  principle  to  these  operations  ;  so  that,  for  example, 

\(I:J)  +  (K-.J)  +  (K:L)\(aJ+  bL)  =  aJ+  (a  +  b)K. 

Finally,  let  us  assume  a  number  of  complex  operations  denoted  by  i',  /,  #, 
I',  etc.,  corresponding  to  the  letters  of  the  algebra  and  determined  by  its  multi 
plication  table  in  the  following  manner  : 

i'=  (I:  A)  +  flll(/:  /)  +  bn(J:  I)  +  cn(K :  I)  +  etc. 

.+  au(I:J)  +  bn(J:  J)  +  Cu(K:  J)  +  etc. 

+  als(/ :  K)  +  bw(J:  K)  +  Cl8(JT:  K)  +  etc.  +  etc. 
/=  (J:A)+  an(I:l)  +  b^(J :  I)  +  %(JT:  /)  +  etc. 

+  0^(1  :J)  +  bn(J:J)  H-  c^(K:J)  +  etc. 

-f  a23(/:  K)  +  bn(J:K)  +  czs(K:  K)  +  etc.  +  etc. 
A/=:  etc. 


*  Any  one  of  them  multiplied  by  0  gives  0 .  t  If  b  =  0 ,  of  course  the  result  is  0 . 


PEIRCB  :    Linear  Associative  Algebra.  127 

Any  two  operations  are  equal  which,  being  performed  on  the  same  operand, 
invariably  give  the  same  result.  The  ultimate  operands  in  this  case  are  the  non- 
relative  units.  But  any  operations  compounded  by  addition  or  multiplication 
of  the  operations  if,  /,  #,  etc.,  if  they  give  the  same  result  when  performed 
upon  A,  will  give  the  same  result  when  performed  upon  any  one  of  the  non- 
relative  units.  For  suppose  i'j'A  =  VI!  A  .  We  have 


i'j'A  =  i'J  =  anl  +  blzJ  +  cnK  +  etc. 

etc. 


so  that  a^  =  aM,  b12  =  b34,  c12  =  c34,  etc.,  and  in  our  original  algebra  ij  =  Id  . 
Hence,  multiplying  both  sides  of  the  equation  into  any  letter,  say  m,  ijm  =  Urn 
But 

ijm  =  i  (a^i  +  bmj  +  c257c  +  etc.)  =  (ana2,  +  a12625  +«13c25  +  etc.)* 

+   (&11«25  +  ^12^25  H-   ^13^25  +   etC.)y  +  GtC. 

But  we  have  equally 

i'j'm'A  =  («na25  H-  a12&25  +  c/13c25  +  etc.)/-f  (An«25  +  /,12625  +  ilst.gB  +  etc.)  J+  etc. 

So  that  i'j'm'A  =  Mm'  A.  Hence,  tyjf  =  MM.  It  follows,  then,  that  if  i'j'A 
=  &W,  then  *'/  into  any  non-relative  unit  equals  M  into  the  same  unit,  so  that 
i'f=  Ml'.  We  thus  see  that  whatever  equality  subsists  between  compounds  of 
the  accented  letters  i',  /,  //,  etc.,  subsists  between  the  same  compounds  of  the 
corresponding  unaccented  letters  i,j,  k,  so  that  the  multiplication  tables  of  the 
two  algebras  are  the  same.*  Thus,  what  has  been  proved  is  that  any  associ- 
tive  algebra  can  be  put  into  relative  form,  *.  e.  (see  my  brochure  entitled 
A  brief  Desertion  of  the  Algebra  of  Relatives)  that  every  such  algebra  may  be 
represented  by  a  matrix. 

Take,  for  example,  the  algebra  (bd<5).     It  takes  the  relative  form 

j  =  (J  :  A)  , 


l=(L:A)  +  (J:K),     m  =  ( M :  A)  +  (r2  -  -  1)  (J :  I)  -  (L  :  K)  -  r2  (J :  M) . 

*  A  brief  proof  of  this  theorem,  perhaps  essentially  the  same  as  the  above,  was  published  by  me  in 
the  Proceedings  of  the  American  Academy  of  Arts  and  Sciences,  for  May  11.  1875. 


128 


PEIRCE:    Linear  Associative  Algebra. 


This  is  the  same  as  to  say  that  the  general  expression  xi  +  yj  +  zk 
of  this  algebra  has  the  same  laws  of  multiplication  as  the  matrix 


o, 

0, 

o, 

o, 

o, 

o, 

X, 

o, 

o, 

2, 

o, 

o, 

X  -|-  2 

lh 

_i_  /«2  |\,y 

o, 

u, 

V2, 

2 

2, 

^  o, 

o, 

0. 

o, 

o, 

U, 

rz, 

o, 

cc  —  v  , 

o, 

—  1 

V, 

0, 

o, 

2, 

o, 

0. 

Of  course,  every  algebra  may  be  put  into  relative  form  in  an  infinity  of 
ways  ;  and  simpler  ways  than  that  which  the  rule  affords  can  often  be  found. 
Thus,  for  the  above  algebra,  the  form  given  in  the  foot-note  is  simpler,  and  so  is 
the  following : 

i=(B:A)  +  (C:B)  +  (F:D)  +  (CiE),    j=(C:A), 
k  =  (2):A)  +  (E:D)+  (C:B)  +  r(F:B)  +  r(C :  F), 
l=(]T.A)  +  (C:D),    m  =  (E:A)  +  (V*—l)(C:E)—(B\A)  —  (F:D}—(C:E). 

These    different  forms  will  suggest  transformations  of  the  algebra.     Thus,  the 
relative  form  in  the  foot-note  to  (H)  suggests  putting 

m  —  —  m  j 


when  we  get  the  following  multiplication  table,  where  p  is  put  for  r 

i          j          Jc         I         m 


0 

0 

0 

0 

j 

0 

0 

0 

0 

0 

0 

0 

i 

i 

1 

0 

0 

9J 

0 

0 

fr 

0 

9l 

0 

.; 

PEIRCE  :    Linear  Associative  Algebra.  129 

Ordinary  algebra  with  imaginaries,  considered  as  a  double  algebra,  is,   in 
relative  form, 

1=(X:X)  +  (Y:  F),      J  =  (X:  Y)-(Y:X). 

This  shows  how  the  operation   J  turns  a  vector  through  a  right  angle  in  the 
plane  of  X,    Y.     Quaternions  in  relative  form  is 

l=.(W:  W)  +  (X:X)  +  (Y:  Y]  +  (Z  :  Z)  , 
i=(X:  W)  --(W:X)  +  (Z:  Y)-(Y:Z), 
j=(Y:  W)  --(Z:X)  -  -  (W  :  Y)  +  (X:Z], 


We  see  that  we  have  here  a  reference  to  a  space  of  four  dimensions  corres 
ponding  to  X,  F,  Z,  W. 

III. 

On  the  Algebras  in  which  Division  is  Unambiguous. 
BY  C.  S.  PEIRCE. 

1.  In  the  Linear  Associative  Algebra,    the  coefficients  are  permitted  to  be 
imaginary.     In  this  note  they  are  restricted  to  being  real.     It  is  assumed  that 
we  have  to  deal  with  an  algebra  such  that  from  AB  =  A  C  we  can  infer  that 
A  =  0  or  B  =  G  .     It  is  required  to  find  what  forms  such  an  algebra  may  take. 

2.  If  AB  =  0,  then  either  A  =  0  or  5  =  0.    For  if  not,  AC=A(B+C), 
although  A  does  not  vanish  and  G  is  unequal  to  B  +  C  . 

3.  The  reasoning  of  §  40  holds,  although  the  coefficients  are  restricted  to 
being    real.      It    is    true,    then,    that    since    there    is    no    expression    (in    the 
algebra  under  consideration)  whose  square  vanishes,  there  must  be  an  expression, 
i,  such  that  i2  =  i. 

4.  By  §  41,  it  appears  that  for  every  expression  in  the  algebra  we  have 

iA  =  Ai  =  A  . 

5.  By  the  reasoning  of  §53,  it  appears  that  for  every  expression  A  there  is 

an  equation  of  the  form 

Zm(amAm)+bi  =  0. 

But  i  is  virtually  arithmetical  unity,  since  iA  =  Ai  =  A  ;  and  this  equation  may 
be  treated  by  the  ordinary  theory  of  equations.  Suppose  it  has  a  real  root,  a  ; 
then  it  will  be  divisible  by  (A  —  a)  ,  and  calling  the  quotient  B  we  shall  have 

(A  —  ai)B  =  0. 


130  P EI ROE  :    Linear  Associative  Algebra. 

But  A  —  ai  is  not  zero,  for  A  was  supposed  dissimilar  to  i  .  Hence  a  product  of 
finites  vanishes,  which  is  impossible.  Hence  the  equation  cannot  have  a  real 
root.  But  the  whole  equation  can  be  resolved  into  quadratic  factors,  and  some 
one  of  these  must  vanish.  Let  the  irresoluble  vanishing  factor  be 

(A  —  -sf  +  *2  — 0. 
Then 


or,  every  expression,  upon  subtraction  of  a  real  number  (i.  e.  a  real  multiple  of  i). 
can  be  converted,  in  one  way  only,  into  a  quantity  whose  square  is  a  negative 
number.  We  may  express  this  by  saying  that  every  quantity  consists  of  a  scalar 
and  a  vector  part.  A  quantity  whose  square  is  a  negative  number  we  here  call 
a  vector. 

6.  Our  next  step  is  to  show  that  the  vector  part  of  the  product  of  two 
vectors  is  linearly  independent  of  these  vectors  and  of  unity.  That  is,  i  and  j 
being  any  two  vectors.*  if 

ij  —  s  -\-  v 

where  s  is  a  scalar  and  v  a  vector,  we  cannot  determine  three  real  scalars 
a,  b,  c,  such  that 

v  =  a  -f-  hi  -f-  cj . 

This  is  proved,  if  we  prove  that  no  scalar  subtracted  from  ij  leaves  a  remainder 
bi  -|-  cj .  If  this  be  true  when  i  and  j  are  any  unit  vectors  whatever,  it  is  true 
when  these  are  multiplied  by  real  scalars,  and  so  is  true  of  every  pair  of  vectors. 
We  will,  then,  suppose  i  and  j  to  be  unit  vectors.  Now, 

ijz  =  —  i. 
If  therefore  we  had 

ij  =  a  +  bi  +  cj , 
we  should  have 

-  i  =.  if  =  aj  -\-  bij  —  c  =  ab  —  c  +  bzi  -\-  (a  +  bc)j ; 
whence,  i  andy  being  dissimilar, 

-i  =  Vi,         b*  =  —  l, 

and  b  could  not  be  real. 


*  The  idempotent  basis  having  been  shown  to  be  arithmetical  unity,  we  are  free  to  use  the  letter  i  to 
denote  another  unit. 


PEIRCE  :    Linear  Associative  Algebra.  131 

7.  Our  next  step  is  to  show  that,  i  and/  being  any  two  vectors,  and 

ij  —  s  +  v, 
s  being  a  scalar  and  v  a  vector,  we  have 

ji  =  r(s  —  v), 

where  r  is  a  real  scalar.     It  will  be  obviously  sufficient  to  prove  this  for  the  case 
in  which  i  and/  are  unit  vectors.     Assuming  them  such,  let  us  write 

ji  =  s'+  v1  ,          vv'—  s"+v"  , 
where  s'  and  s"  are  scalars,  while  v'  and  v"  are  vectors.     Then 

ij.ji  —  (-S-  +  V)  (.S-'-f  V')  =  6V/  +  SV+  S'V  +  V"+  S". 


But  we  have 

ij  .ji  •=.  ij*i  —  —  p  —  1  . 

Hence, 

v"=  1  —  ss1—  s"—  *v'—  s'v. 

But  v"  is  the  vector  of  vv',  so  that  by  the  last  paragraph  such  an  equation  cannot 
subsist  unless  v"  vanishes.     Thus  we  get 

0=1—  ss'—  s"—  sv'—  s'v  , 
or 

8V'=  1  -  SS1  -  s"  -  s'v  . 

But  a  quantity  can  only  be  separated  in  one  way  into  a  scalar  and  a  vector  part  ; 
so  that 


That  is, 

ji=~(s-v).     Q.E.D. 

S 

8.  Our  next  step  is  to  prove  that  s  =  s';  so  that  if  ij  =  s  H-  v  then  ji  = 
s  —  v.  It  is  obviously  sufficient  to  prove  this  when  i  and  /  are  unit  vectors.  Now 
from  any  quantity  a  scalar  may  be  subtracted  so  as  to  leave  a  remainder  whose 
square  is  a  scalar.  We  do  not  yet  know  whether  the  sum  of  two  vectors  is  a 
vector  or  not  (though  we  do  know  that  it  is  not  a  scalar).  Let  us  then  take  such 
a  sum  as  ai  +  bj  and  suppose  x  to  be  the  scalar  which  subtracted  from  it  makes 
the  square  of  the  remainder  a  scalar.  Then,  C  being  a  scalar, 

—»  +  «»  +      »=  C. 


132  PEIRCE  :    Linear  Associative  Algebra. 

But  developing  the  square  we  have 

(—  x  +  ai  +  bjj  =  xz  —  a*  —  I?  +  abs  +  abs1  —  2axi  +  2bxj  +  ab(l  —  -  \=  C; 


V 
i.  e. 

ab(l  —  -}v  =  C—x2  +  a2  +  tf  —  abs  —  abs'  +  2cm  +  2bxj . 

\  9  / 

But  v  being  the  vector  of  i/,  by  the  last  paragraph  but  one  the  equation  must 
vanish.  Either  then  v  =  0  or  1  -  ' --  =  0  .  But  if  v  =  0 ,  ij  =  s,  and  multiply 
ing  into/, 

~~"     v     *  Oj    * 

at 

which  is  absurd,  i  and  /  being  dissimilar.     Hence  1 =  0  and 

s 

ji  —  s  —  v.      Q.E.D. 

9.  The  number  of  independent  vectors  in  the  algebra  cannot  be  two.  For 
the  vector  of  ij  is  independent  of  i  and/.  There  may  be  no  vector,  and  in  that 
case  we  have  the  ordinary  algebra  of  reals ;  or  there  may  be  only  one  vector, 
and  in  that  case  we  have  the  ordinary  algebra  of  imaginaries. 

Let  i  and/  be  two  independent  vectors  such  that 

ij  =•  s  -\-  v . 
Let  us  substitute  for  / 

/!  =  si  +  / . 
Then  we  have 


Jv  =   ii  =  —       =    ,     vj  =      =  —, 
iv  =  <fy\  =  —j\ ,     vi  =  ij\i  =  —  j$  =  /! . 

Thus  we  have  the  algebra  of  real  quaternions.     Suppose  we  have  a  fourth   unit 
vector,  Jc ,  linearly  independent  of  all  the  others,  and  let  us  write   ' 

jjc  =  s1  +  V*, 

ld  =  s"+  v". 
Let  us  substitute  for  k 

IfcSSjft  +  <£  +  *! 

and  we  get 

JA  =  —s"v  +  tf,     k^\  =  s"v  —  tf, 
kfi  =  —  s'v  +  v",       ik^  =  s'v  —  v". 


PEIRCE  :    Linear  Associative  Algebra.  133 

Let  us  further  suppose 

(itik,  =  s>  +  »"'. 
Then,  because  ij\  is  a  vector, 

kl(ij\)  =  Sf"-vfff. 
But 

hji  =  —JA ,     kj<  —  —  iki , 
because  both  products  are  vectors. 

Hence 

*  -JA  =  —i.  &i/i  =  —  MI  -ji  =  %  -ji  =  &i  •  ij\  • 
Hence 

s"'+  v"'=  s"'—v"' 

or  v"'=  0 ,  and  the  product  of  the  two  unit  vectors  is  a  scalar.  These  vectors 
cannot,  then,  be  independent^  or  k  cannot  be  independent  of  if  =  v .  Thus  it  is 
proved  that  a  fourth  independent  vector  is  impossible,  and  that  ordinary  real 
algebra,  ordinary  algebra  with  irnaginaries,  and  real  quaternions  are  the  only 
associative  algebras  in  which  division  by  finites  always  yields  an  unambiguous 
quotient. 


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